| L(s) = 1 | − 3·2-s + 4·4-s − 5-s + 2·7-s − 3·8-s + 3·10-s + 2·13-s − 6·14-s + 3·16-s + 6·17-s + 8·19-s − 4·20-s + 10·23-s − 8·25-s − 6·26-s + 8·28-s − 6·29-s − 4·31-s − 6·32-s − 18·34-s − 2·35-s − 8·37-s − 24·38-s + 3·40-s − 5·41-s − 18·43-s − 30·46-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s − 0.447·5-s + 0.755·7-s − 1.06·8-s + 0.948·10-s + 0.554·13-s − 1.60·14-s + 3/4·16-s + 1.45·17-s + 1.83·19-s − 0.894·20-s + 2.08·23-s − 8/5·25-s − 1.17·26-s + 1.51·28-s − 1.11·29-s − 0.718·31-s − 1.06·32-s − 3.08·34-s − 0.338·35-s − 1.31·37-s − 3.89·38-s + 0.474·40-s − 0.780·41-s − 2.74·43-s − 4.42·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 83 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 23 T + 249 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 77 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 237 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 135 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 159 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 227 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.62982613650095740300013889362, −7.57367151838280909557032402807, −7.26546680434960463772068067742, −7.10744698461913693868347330266, −6.43185825806961427504287753968, −5.98363814069064791399462479166, −5.72896813245293297572814408007, −5.16065632043477594080739245144, −4.99857193671052777926320270104, −4.76234871193544451879376802813, −3.88434293570584347976550133206, −3.45226789280995169202753723838, −3.25938163883079980814835395062, −3.11840836601130010829900086031, −2.08287091161085262958544828040, −1.64987671823058473048000763553, −1.20738049757599581599665986864, −1.18884260126167992936749636724, 0, 0,
1.18884260126167992936749636724, 1.20738049757599581599665986864, 1.64987671823058473048000763553, 2.08287091161085262958544828040, 3.11840836601130010829900086031, 3.25938163883079980814835395062, 3.45226789280995169202753723838, 3.88434293570584347976550133206, 4.76234871193544451879376802813, 4.99857193671052777926320270104, 5.16065632043477594080739245144, 5.72896813245293297572814408007, 5.98363814069064791399462479166, 6.43185825806961427504287753968, 7.10744698461913693868347330266, 7.26546680434960463772068067742, 7.57367151838280909557032402807, 7.62982613650095740300013889362
