| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 6·11-s − 12-s + 2·13-s + 16-s + 18-s − 6·19-s + 6·22-s + 4·23-s − 24-s + 2·26-s − 27-s + 4·31-s + 32-s − 6·33-s + 36-s − 6·37-s − 6·38-s − 2·39-s − 6·41-s − 4·43-s + 6·44-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 + 1.80·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.235·18-s − 1.37·19-s + 1.27·22-s + 0.834·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.718·31-s + 0.176·32-s − 1.04·33-s + 1/6·36-s − 0.986·37-s − 0.973·38-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 53 | \( 1 + T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.49462443831848, −12.32267026982850, −11.75822441082362, −11.48644003594989, −10.93520129571196, −10.62774007749329, −10.12099172932474, −9.518141575437106, −9.078647439367343, −8.528917002530788, −8.279972082614490, −7.428960015664720, −6.878254990891911, −6.667594327426931, −6.188965407467268, −5.873642651985088, −5.129160750836313, −4.705205391548790, −4.255109921769313, −3.740904910213166, −3.374199672246761, −2.673668655657246, −1.864951153180601, −1.489386892299667, −0.9030257096757960, 0,
0.9030257096757960, 1.489386892299667, 1.864951153180601, 2.673668655657246, 3.374199672246761, 3.740904910213166, 4.255109921769313, 4.705205391548790, 5.129160750836313, 5.873642651985088, 6.188965407467268, 6.667594327426931, 6.878254990891911, 7.428960015664720, 8.279972082614490, 8.528917002530788, 9.078647439367343, 9.518141575437106, 10.12099172932474, 10.62774007749329, 10.93520129571196, 11.48644003594989, 11.75822441082362, 12.32267026982850, 12.49462443831848
