| L(s) = 1 | + 3-s + 7-s + 9-s + 2·13-s − 2·17-s − 2·19-s + 21-s + 2·23-s − 5·25-s + 27-s + 10·29-s + 4·31-s − 2·37-s + 2·39-s + 2·41-s + 6·43-s + 10·47-s + 49-s − 2·51-s + 12·53-s − 2·57-s − 4·59-s + 2·61-s + 63-s + 12·67-s + 2·69-s − 2·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.485·17-s − 0.458·19-s + 0.218·21-s + 0.417·23-s − 25-s + 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.328·37-s + 0.320·39-s + 0.312·41-s + 0.914·43-s + 1.45·47-s + 1/7·49-s − 0.280·51-s + 1.64·53-s − 0.264·57-s − 0.520·59-s + 0.256·61-s + 0.125·63-s + 1.46·67-s + 0.240·69-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.555283325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.555283325\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.76061802384735, −14.14517300125695, −13.70226667423496, −13.48228874430092, −12.60008086782542, −12.30308140333365, −11.60576981566191, −11.11332561651850, −10.47077833520412, −10.12302851992739, −9.399132977998816, −8.754669833837237, −8.509980571920273, −7.878462363941057, −7.311342642600960, −6.650442408959472, −6.160859689156908, −5.438651611728778, −4.751722424365520, −4.104105189869509, −3.725166743082867, −2.636240322411504, −2.417847080877761, −1.401615794352109, −0.6996032502216066,
0.6996032502216066, 1.401615794352109, 2.417847080877761, 2.636240322411504, 3.725166743082867, 4.104105189869509, 4.751722424365520, 5.438651611728778, 6.160859689156908, 6.650442408959472, 7.311342642600960, 7.878462363941057, 8.509980571920273, 8.754669833837237, 9.399132977998816, 10.12302851992739, 10.47077833520412, 11.11332561651850, 11.60576981566191, 12.30308140333365, 12.60008086782542, 13.48228874430092, 13.70226667423496, 14.14517300125695, 14.76061802384735
