| L(s) = 1 | + 2·2-s − 4-s + 16·7-s + 4·8-s + 22·11-s − 80·13-s + 32·14-s − 19·16-s + 164·17-s − 36·19-s + 44·22-s + 172·23-s − 160·26-s − 16·28-s − 108·29-s − 448·31-s − 202·32-s + 328·34-s − 108·37-s − 72·38-s − 212·41-s − 156·43-s − 22·44-s + 344·46-s − 20·47-s − 194·49-s + 80·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/8·4-s + 0.863·7-s + 0.176·8-s + 0.603·11-s − 1.70·13-s + 0.610·14-s − 0.296·16-s + 2.33·17-s − 0.434·19-s + 0.426·22-s + 1.55·23-s − 1.20·26-s − 0.107·28-s − 0.691·29-s − 2.59·31-s − 1.11·32-s + 1.65·34-s − 0.479·37-s − 0.307·38-s − 0.807·41-s − 0.553·43-s − 0.0753·44-s + 1.10·46-s − 0.0620·47-s − 0.565·49-s + 0.213·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 5 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 450 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4542 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 164 T + 16118 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 36 T + 3242 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 172 T + 30758 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 108 T + 14062 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 448 T + 101646 T^{2} + 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 108 T + 103022 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 212 T + 148886 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 156 T + 63530 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 20 T + 202454 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 132 T - 117518 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 688 T + 404246 T^{2} + 688 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 96 T + 402398 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 448 T + 455094 T^{2} + 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 132 T - 187322 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 428 T + 808278 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 424 T + 1031010 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 720 T + 1262374 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1056 T + 1317010 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 52 T - 413466 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.105178168344198482354032114904, −8.049908114061790220862819089969, −7.58506472900526301457368252198, −7.23344725189464326407376609754, −6.85874865580897691595800599802, −6.66726644961593063064459442026, −5.63255918478702232220884879074, −5.60709394848988254923643787746, −5.07613766603473513544177414548, −5.07142664236216454286872069045, −4.36228566246895386629161448926, −4.12899786634204469406910749871, −3.46457230935938340372834519793, −3.19809142415268136166474288638, −2.69139099192013322976837529134, −1.86557559747747537582202254754, −1.58929772862280882963742224878, −1.19297717262362627579429481781, 0, 0,
1.19297717262362627579429481781, 1.58929772862280882963742224878, 1.86557559747747537582202254754, 2.69139099192013322976837529134, 3.19809142415268136166474288638, 3.46457230935938340372834519793, 4.12899786634204469406910749871, 4.36228566246895386629161448926, 5.07142664236216454286872069045, 5.07613766603473513544177414548, 5.60709394848988254923643787746, 5.63255918478702232220884879074, 6.66726644961593063064459442026, 6.85874865580897691595800599802, 7.23344725189464326407376609754, 7.58506472900526301457368252198, 8.049908114061790220862819089969, 8.105178168344198482354032114904
