| L(s) = 1 | − 3-s − 4·5-s + 4·7-s + 9-s + 2·11-s − 13-s + 4·15-s − 6·17-s − 4·19-s − 4·21-s − 4·23-s + 11·25-s − 27-s − 6·29-s − 8·31-s − 2·33-s − 16·35-s − 10·37-s + 39-s − 4·41-s + 4·43-s − 4·45-s + 6·47-s + 9·49-s + 6·51-s + 6·53-s − 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 1.03·15-s − 1.45·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s + 11/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.348·33-s − 2.70·35-s − 1.64·37-s + 0.160·39-s − 0.624·41-s + 0.609·43-s − 0.596·45-s + 0.875·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−10.82355188850108025239299977249, −9.011032696371544011165812270443, −8.370619656211679532178861176112, −7.49238534345418766392709628204, −6.84343385941693059687900536607, −5.38848200883445772598444752149, −4.36649745619438974292890469655, −3.91612461683650343655243869043, −1.88069027517331870558643695148, 0,
1.88069027517331870558643695148, 3.91612461683650343655243869043, 4.36649745619438974292890469655, 5.38848200883445772598444752149, 6.84343385941693059687900536607, 7.49238534345418766392709628204, 8.370619656211679532178861176112, 9.011032696371544011165812270443, 10.82355188850108025239299977249
