| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 2·11-s − 12-s − 3·13-s − 14-s + 16-s − 2·17-s − 18-s + 19-s − 21-s + 2·22-s + 4·23-s + 24-s + 3·26-s − 27-s + 28-s − 2·29-s + 8·31-s − 32-s + 2·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.218·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.348·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 43 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−12.84602970586820, −12.50515240394570, −11.85798449848014, −11.63054218159516, −11.02597332374647, −10.76824361145053, −10.18700553827585, −9.849543325470354, −9.322401688168850, −8.879626643195757, −8.295760118974914, −7.814545155922269, −7.464224241335388, −6.962638590599613, −6.359311195691301, −6.104807424934491, −5.191122048709905, −4.999669474092255, −4.564443012583792, −3.737347720052282, −3.139871358934579, −2.459576643090463, −2.093113014531442, −1.278857551103565, −0.6976590926813208, 0,
0.6976590926813208, 1.278857551103565, 2.093113014531442, 2.459576643090463, 3.139871358934579, 3.737347720052282, 4.564443012583792, 4.999669474092255, 5.191122048709905, 6.104807424934491, 6.359311195691301, 6.962638590599613, 7.464224241335388, 7.814545155922269, 8.295760118974914, 8.879626643195757, 9.322401688168850, 9.849543325470354, 10.18700553827585, 10.76824361145053, 11.02597332374647, 11.63054218159516, 11.85798449848014, 12.50515240394570, 12.84602970586820
