| L(s) = 1 | − 4-s + 10·7-s + 10·13-s − 16-s + 2·19-s − 10·28-s − 6·31-s + 50·37-s + 29·49-s − 10·52-s − 54·61-s + 10·67-s + 30·73-s − 2·76-s + 28·79-s + 100·91-s + 70·103-s − 12·109-s − 10·112-s − 28·121-s + 6·124-s + 127-s + 131-s + 20·133-s + 137-s + 139-s − 50·148-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 3.77·7-s + 2.77·13-s − 1/4·16-s + 0.458·19-s − 1.88·28-s − 1.07·31-s + 8.21·37-s + 29/7·49-s − 1.38·52-s − 6.91·61-s + 1.22·67-s + 3.51·73-s − 0.229·76-s + 3.15·79-s + 10.4·91-s + 6.89·103-s − 1.14·109-s − 0.944·112-s − 2.54·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 1.73·133-s + 0.0854·137-s + 0.0848·139-s − 4.10·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(61.58409792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(61.58409792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + T^{2} + p T^{4} + 3 T^{6} + 25 T^{8} + 3 p^{2} T^{10} + p^{5} T^{12} + p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 5 T + 23 T^{2} - 75 T^{3} + 244 T^{4} - 75 p T^{5} + 23 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 28 T^{2} + 503 T^{4} + 5436 T^{6} + 61120 T^{8} + 5436 p^{2} T^{10} + 503 p^{4} T^{12} + 28 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 5 T + 42 T^{2} - 160 T^{3} + 749 T^{4} - 160 p T^{5} + 42 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 36 T^{2} + 1007 T^{4} + 19188 T^{6} + 344160 T^{8} + 19188 p^{2} T^{10} + 1007 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - T + 62 T^{2} - 48 T^{3} + 1675 T^{4} - 48 p T^{5} + 62 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 19 T^{2} + 997 T^{4} + 14497 T^{6} + 447820 T^{8} + 14497 p^{2} T^{10} + 997 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 + 97 T^{2} + 4733 T^{4} + 181059 T^{6} + 5875420 T^{8} + 181059 p^{2} T^{10} + 4733 p^{4} T^{12} + 97 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 3 T + 108 T^{2} + 266 T^{3} + 4815 T^{4} + 266 p T^{5} + 108 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 25 T + 363 T^{2} - 3495 T^{3} + 24844 T^{4} - 3495 p T^{5} + 363 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 43 T^{2} + 1133 T^{4} + 60501 T^{6} + 1445620 T^{8} + 60501 p^{2} T^{10} + 1133 p^{4} T^{12} + 43 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 97 T^{2} + 5389 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 276 T^{2} + 35732 T^{4} + 2879148 T^{6} + 160338390 T^{8} + 2879148 p^{2} T^{10} + 35732 p^{4} T^{12} + 276 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 244 T^{2} + 25412 T^{4} + 1584492 T^{6} + 82074070 T^{8} + 1584492 p^{2} T^{10} + 25412 p^{4} T^{12} + 244 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 + 172 T^{2} + 24743 T^{4} + 2088324 T^{6} + 150835120 T^{8} + 2088324 p^{2} T^{10} + 24743 p^{4} T^{12} + 172 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 27 T + 498 T^{2} + 5924 T^{3} + 54645 T^{4} + 5924 p T^{5} + 498 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 5 T + 108 T^{2} - 460 T^{3} + 8789 T^{4} - 460 p T^{5} + 108 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 273 T^{2} + 41753 T^{4} + 4454691 T^{6} + 360816420 T^{8} + 4454691 p^{2} T^{10} + 41753 p^{4} T^{12} + 273 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 15 T + 207 T^{2} - 1145 T^{3} + 11484 T^{4} - 1145 p T^{5} + 207 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 14 T + 297 T^{2} - 2682 T^{3} + 33535 T^{4} - 2682 p T^{5} + 297 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 244 T^{2} + 43127 T^{4} + 5300892 T^{6} + 498372400 T^{8} + 5300892 p^{2} T^{10} + 43127 p^{4} T^{12} + 244 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 112 T^{2} + 18263 T^{4} + 2446764 T^{6} + 185830120 T^{8} + 2446764 p^{2} T^{10} + 18263 p^{4} T^{12} + 112 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 343 T^{2} - 90 T^{3} + 47719 T^{4} - 90 p T^{5} + 343 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.31966001931709322323146250682, −3.20920672122029436848582925096, −3.19746275324113116074956192238, −2.97149539164991883928776863529, −2.81585365397507598914764965500, −2.70842297550166977965470621999, −2.61781070522780820660846198255, −2.51142708551713243934303710408, −2.32307936659330409320205869754, −2.30938110422499450208540543409, −2.25680319654246704094234343203, −1.87585142182588540528248479671, −1.87335251676570113231403340523, −1.83878036191069945080091533036, −1.56685716945302925438161965488, −1.56520218934925892111906316458, −1.42744269481892478503903078877, −1.35929699446451392608734575075, −1.16841736352817148927343640594, −0.948722355471165108977415426575, −0.912425096168884309632086098525, −0.72359189192928754695304288969, −0.60682765489845672991270010673, −0.53026171254429675352790451848, −0.26101456449982568424260601625,
0.26101456449982568424260601625, 0.53026171254429675352790451848, 0.60682765489845672991270010673, 0.72359189192928754695304288969, 0.912425096168884309632086098525, 0.948722355471165108977415426575, 1.16841736352817148927343640594, 1.35929699446451392608734575075, 1.42744269481892478503903078877, 1.56520218934925892111906316458, 1.56685716945302925438161965488, 1.83878036191069945080091533036, 1.87335251676570113231403340523, 1.87585142182588540528248479671, 2.25680319654246704094234343203, 2.30938110422499450208540543409, 2.32307936659330409320205869754, 2.51142708551713243934303710408, 2.61781070522780820660846198255, 2.70842297550166977965470621999, 2.81585365397507598914764965500, 2.97149539164991883928776863529, 3.19746275324113116074956192238, 3.20920672122029436848582925096, 3.31966001931709322323146250682
Plot not available for L-functions of degree greater than 10.
