| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 4·11-s + 12-s + 4·13-s + 16-s − 2·17-s + 18-s + 4·22-s − 4·23-s + 24-s + 4·26-s + 27-s − 4·31-s + 32-s + 4·33-s − 2·34-s + 36-s − 2·37-s + 4·39-s + 41-s + 12·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.852·22-s − 0.834·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.718·31-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s − 0.328·37-s + 0.640·39-s + 0.156·41-s + 1.82·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−12.86881270045218, −12.57318054715211, −12.08035458332599, −11.53117222281115, −11.21008949412587, −10.71421151054645, −10.25622797154710, −9.578030085156803, −9.247518155145546, −8.716197886244775, −8.356317592845599, −7.816889417611079, −7.110405940488467, −6.917933454012092, −6.287511933405900, −5.758586336688692, −5.551325888454041, −4.538534247290507, −4.147080082905294, −3.940558608493859, −3.291288059016383, −2.761302570547050, −2.123064432457077, −1.489208389663820, −1.119097697615468, 0,
1.119097697615468, 1.489208389663820, 2.123064432457077, 2.761302570547050, 3.291288059016383, 3.940558608493859, 4.147080082905294, 4.538534247290507, 5.551325888454041, 5.758586336688692, 6.287511933405900, 6.917933454012092, 7.110405940488467, 7.816889417611079, 8.356317592845599, 8.716197886244775, 9.247518155145546, 9.578030085156803, 10.25622797154710, 10.71421151054645, 11.21008949412587, 11.53117222281115, 12.08035458332599, 12.57318054715211, 12.86881270045218
