| L(s) = 1 | + 3-s + 4·7-s + 9-s − 6·11-s + 13-s + 2·17-s + 4·21-s − 5·25-s + 27-s + 2·29-s − 8·31-s − 6·33-s + 2·37-s + 39-s + 4·41-s − 4·43-s + 6·47-s + 9·49-s + 2·51-s − 2·53-s − 10·59-s + 10·61-s + 4·63-s − 8·67-s − 2·71-s + 14·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 0.485·17-s + 0.872·21-s − 25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 1.04·33-s + 0.328·37-s + 0.160·39-s + 0.624·41-s − 0.609·43-s + 0.875·47-s + 9/7·49-s + 0.280·51-s − 0.274·53-s − 1.30·59-s + 1.28·61-s + 0.503·63-s − 0.977·67-s − 0.237·71-s + 1.63·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.175829200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.175829200\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.78229883485416, −13.27270996750153, −12.62408044569220, −12.31730293031605, −11.62674753294579, −11.03433893240928, −10.81780712451003, −10.28186543889110, −9.713824531037932, −9.161839792805829, −8.559868758799357, −8.033796504172817, −7.804284579037224, −7.457037029847644, −6.749227791391637, −5.788157090385317, −5.554508954833030, −4.911993607738998, −4.515590060795886, −3.773586870922466, −3.218992492493268, −2.421626771446997, −2.065411965097708, −1.403120618209803, −0.5183732030047187,
0.5183732030047187, 1.403120618209803, 2.065411965097708, 2.421626771446997, 3.218992492493268, 3.773586870922466, 4.515590060795886, 4.911993607738998, 5.554508954833030, 5.788157090385317, 6.749227791391637, 7.457037029847644, 7.804284579037224, 8.033796504172817, 8.559868758799357, 9.161839792805829, 9.713824531037932, 10.28186543889110, 10.81780712451003, 11.03433893240928, 11.62674753294579, 12.31730293031605, 12.62408044569220, 13.27270996750153, 13.78229883485416
