| L(s) = 1 | − 3-s − 2·4-s + 7-s + 9-s − 2·11-s + 2·12-s − 13-s + 4·16-s − 4·19-s − 21-s − 4·23-s − 27-s − 2·28-s − 8·29-s + 2·33-s − 2·36-s − 3·37-s + 39-s + 6·41-s + 8·43-s + 4·44-s − 6·47-s − 4·48-s − 6·49-s + 2·52-s − 10·53-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 0.277·13-s + 16-s − 0.917·19-s − 0.218·21-s − 0.834·23-s − 0.192·27-s − 0.377·28-s − 1.48·29-s + 0.348·33-s − 1/3·36-s − 0.493·37-s + 0.160·39-s + 0.937·41-s + 1.21·43-s + 0.603·44-s − 0.875·47-s − 0.577·48-s − 6/7·49-s + 0.277·52-s − 1.37·53-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.93890871191075, −12.54800637196863, −12.18445770024644, −11.62158530871489, −10.99649073565352, −10.68297309399928, −10.36381034429015, −9.623608219516956, −9.318783716766904, −8.984930313487925, −8.165978567006046, −7.842998781439163, −7.624293238676815, −6.831144157525031, −6.167920068317849, −5.876513049227170, −5.327585032276014, −4.778586590722342, −4.473554726199609, −3.947470988380912, −3.348362928927160, −2.701482594410351, −1.853763052641499, −1.513720431016438, −0.4884526492809938, 0,
0.4884526492809938, 1.513720431016438, 1.853763052641499, 2.701482594410351, 3.348362928927160, 3.947470988380912, 4.473554726199609, 4.778586590722342, 5.327585032276014, 5.876513049227170, 6.167920068317849, 6.831144157525031, 7.624293238676815, 7.842998781439163, 8.165978567006046, 8.984930313487925, 9.318783716766904, 9.623608219516956, 10.36381034429015, 10.68297309399928, 10.99649073565352, 11.62158530871489, 12.18445770024644, 12.54800637196863, 12.93890871191075
