| L(s) = 1 | + 4·5-s − 14·7-s − 48·11-s − 52·13-s + 12·17-s + 160·19-s − 48·23-s − 90·25-s + 380·29-s + 184·31-s − 56·35-s + 84·37-s − 52·41-s − 384·43-s − 64·47-s + 147·49-s + 652·53-s − 192·55-s − 424·59-s − 1.06e3·61-s − 208·65-s − 1.31e3·67-s − 672·71-s − 1.08e3·73-s + 672·77-s − 736·79-s + 280·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s − 0.755·7-s − 1.31·11-s − 1.10·13-s + 0.171·17-s + 1.93·19-s − 0.435·23-s − 0.719·25-s + 2.43·29-s + 1.06·31-s − 0.270·35-s + 0.373·37-s − 0.198·41-s − 1.36·43-s − 0.198·47-s + 3/7·49-s + 1.68·53-s − 0.470·55-s − 0.935·59-s − 2.22·61-s − 0.396·65-s − 2.39·67-s − 1.12·71-s − 1.73·73-s + 0.994·77-s − 1.04·79-s + 0.370·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 106 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 48 T + 3090 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 p T + 2702 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 2610 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 160 T + 19526 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 48 T + 21210 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 380 T + 79550 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 184 T + 20094 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 84 T + 43870 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 52 T + 3346 p T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 384 T + 157990 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 64 T - 5042 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 652 T + 212222 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 424 T + 384070 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1060 T + 729534 T^{2} + 1060 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1312 T + 898662 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 672 T + 443770 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1084 T + 1071206 T^{2} + 1084 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 736 T + 907790 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 280 T + 926374 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 948 T + 514402 T^{2} + 948 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 804 T + 1886902 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.672267510910292258730421775788, −8.045447105026146343293563050696, −7.77231460074006979684035485834, −7.54704668609962058185121243263, −7.05806101347779986370244692020, −6.68230807080349005512604110305, −6.01466047789461932118257372293, −5.98255187825241133214641071833, −5.35803380819465512607835531338, −5.02487735303190817650754141672, −4.46116209418531877760785562823, −4.36752969350611414444394010112, −3.24480356236881737822781727399, −3.10355806103462413643630951036, −2.75963262974494864369031005923, −2.29077338661733362818397450615, −1.43872968926350418956866716379, −1.03078611871372511477320328094, 0, 0,
1.03078611871372511477320328094, 1.43872968926350418956866716379, 2.29077338661733362818397450615, 2.75963262974494864369031005923, 3.10355806103462413643630951036, 3.24480356236881737822781727399, 4.36752969350611414444394010112, 4.46116209418531877760785562823, 5.02487735303190817650754141672, 5.35803380819465512607835531338, 5.98255187825241133214641071833, 6.01466047789461932118257372293, 6.68230807080349005512604110305, 7.05806101347779986370244692020, 7.54704668609962058185121243263, 7.77231460074006979684035485834, 8.045447105026146343293563050696, 8.672267510910292258730421775788
