| L(s) = 1 | − 2·3-s − 3·5-s + 9-s + 11-s − 3·13-s + 6·15-s + 2·17-s − 4·19-s + 2·23-s + 4·25-s + 4·27-s − 2·29-s − 31-s − 2·33-s − 37-s + 6·39-s + 6·41-s + 2·43-s − 3·45-s + 12·47-s − 4·51-s − 9·53-s − 3·55-s + 8·57-s − 59-s + 2·61-s + 9·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 1/3·9-s + 0.301·11-s − 0.832·13-s + 1.54·15-s + 0.485·17-s − 0.917·19-s + 0.417·23-s + 4/5·25-s + 0.769·27-s − 0.371·29-s − 0.179·31-s − 0.348·33-s − 0.164·37-s + 0.960·39-s + 0.937·41-s + 0.304·43-s − 0.447·45-s + 1.75·47-s − 0.560·51-s − 1.23·53-s − 0.404·55-s + 1.05·57-s − 0.130·59-s + 0.256·61-s + 1.11·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.99752280760240, −13.07454664581951, −12.63102390008320, −12.33353244074858, −11.87234254277386, −11.42261429959685, −11.11609275010951, −10.51978860915812, −10.22369770828024, −9.376007321763859, −8.950365532522561, −8.374641067214971, −7.735079007915002, −7.390319146651473, −6.903875709856387, −6.285649308524610, −5.764277600238815, −5.274595374312387, −4.531845616412169, −4.333588197171213, −3.640659870600191, −2.975167647962240, −2.314250583920006, −1.335830264058810, −0.5845062422642451, 0,
0.5845062422642451, 1.335830264058810, 2.314250583920006, 2.975167647962240, 3.640659870600191, 4.333588197171213, 4.531845616412169, 5.274595374312387, 5.764277600238815, 6.285649308524610, 6.903875709856387, 7.390319146651473, 7.735079007915002, 8.374641067214971, 8.950365532522561, 9.376007321763859, 10.22369770828024, 10.51978860915812, 11.11609275010951, 11.42261429959685, 11.87234254277386, 12.33353244074858, 12.63102390008320, 13.07454664581951, 13.99752280760240
