| L(s) = 1 | + 7·2-s + 3·3-s + 25·4-s + 21·6-s + 25·7-s + 63·8-s − 43·9-s − 22·11-s + 75·12-s + 50·13-s + 175·14-s + 169·16-s + 151·17-s − 301·18-s − 3·19-s + 75·21-s − 154·22-s − 48·23-s + 189·24-s + 350·26-s − 204·27-s + 625·28-s − 221·29-s + 141·31-s + 623·32-s − 66·33-s + 1.05e3·34-s + ⋯ |
| L(s) = 1 | + 2.47·2-s + 0.577·3-s + 25/8·4-s + 1.42·6-s + 1.34·7-s + 2.78·8-s − 1.59·9-s − 0.603·11-s + 1.80·12-s + 1.06·13-s + 3.34·14-s + 2.64·16-s + 2.15·17-s − 3.94·18-s − 0.0362·19-s + 0.779·21-s − 1.49·22-s − 0.435·23-s + 1.60·24-s + 2.64·26-s − 1.45·27-s + 4.21·28-s − 1.41·29-s + 0.816·31-s + 3.44·32-s − 0.348·33-s + 5.33·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(15.08640241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.08640241\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 7 T + 3 p^{3} T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 52 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 25 T + 498 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 50 T + 4594 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 151 T + 14298 T^{2} - 151 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 5114 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 48 T + 24842 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 221 T + 39564 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 141 T + 6374 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 559 T + 171568 T^{2} - 559 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 144 T + 39598 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 60 T + 56486 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 48 T + 3526 p T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 117 T + 179792 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 86 T + 306510 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 155 T + 399784 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 266 T + 523590 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1587 T + 1329650 T^{2} - 1587 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 70 T + 764962 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1294 T + 1349454 T^{2} + 1294 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 558 T + 911590 T^{2} - 558 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1777 T + 2191512 T^{2} + 1777 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 334 T + 977242 T^{2} - 334 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.69572016605328762640258971878, −11.59803719714915946088371471098, −10.94642779717614699350595196286, −10.73996577671307568473371378291, −9.791117252549633231185960275416, −9.449723296235921756473456115223, −8.346354681001119401122685413277, −8.178086329002355171538578448866, −7.928979957336453317481974079833, −7.29460882652785275914456489596, −6.05652779798234456546893902156, −6.03997994047472442566270334482, −5.34495203944260822115612825842, −5.25152315439754542348016722924, −4.35860796720789923867682379772, −3.93093175354559237603576379166, −3.15864310528863240940297641906, −2.96533231805879633072178280893, −2.03656860698502110908573076507, −1.04069971571058297044101241945,
1.04069971571058297044101241945, 2.03656860698502110908573076507, 2.96533231805879633072178280893, 3.15864310528863240940297641906, 3.93093175354559237603576379166, 4.35860796720789923867682379772, 5.25152315439754542348016722924, 5.34495203944260822115612825842, 6.03997994047472442566270334482, 6.05652779798234456546893902156, 7.29460882652785275914456489596, 7.928979957336453317481974079833, 8.178086329002355171538578448866, 8.346354681001119401122685413277, 9.449723296235921756473456115223, 9.791117252549633231185960275416, 10.73996577671307568473371378291, 10.94642779717614699350595196286, 11.59803719714915946088371471098, 11.69572016605328762640258971878
