| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 2·11-s + 12-s − 4·13-s − 14-s + 16-s − 17-s + 18-s − 2·19-s − 21-s − 2·22-s + 4·23-s + 24-s − 4·26-s + 27-s − 28-s + 32-s − 2·33-s − 34-s + 36-s + 8·37-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 − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.458·19-s − 0.218·21-s − 0.426·22-s + 0.834·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s − 0.188·28-s + 0.176·32-s − 0.348·33-s − 0.171·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.89800758752194, −15.36148136932748, −14.97002896420503, −14.49787716352787, −13.87040561097815, −13.30723129213824, −12.86620410672445, −12.42658831811854, −11.80752113073666, −11.07311241243803, −10.54510993322978, −9.934000319388942, −9.363935169378083, −8.755571517440145, −7.984037615569337, −7.434315803645587, −6.953252585342241, −6.207876203517621, −5.554562023768898, −4.773411915284684, −4.355184130268831, −3.501770266596626, −2.684480814851785, −2.448391188127436, −1.317780528475298, 0,
1.317780528475298, 2.448391188127436, 2.684480814851785, 3.501770266596626, 4.355184130268831, 4.773411915284684, 5.554562023768898, 6.207876203517621, 6.953252585342241, 7.434315803645587, 7.984037615569337, 8.755571517440145, 9.363935169378083, 9.934000319388942, 10.54510993322978, 11.07311241243803, 11.80752113073666, 12.42658831811854, 12.86620410672445, 13.30723129213824, 13.87040561097815, 14.49787716352787, 14.97002896420503, 15.36148136932748, 15.89800758752194
