| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 16-s + 2·17-s − 2·19-s − 20-s + 22-s + 6·23-s + 25-s + 2·29-s − 32-s − 2·34-s + 6·37-s + 2·38-s + 40-s − 2·41-s − 2·43-s − 44-s − 6·46-s + 2·47-s − 7·49-s − 50-s + 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.176·32-s − 0.342·34-s + 0.986·37-s + 0.324·38-s + 0.158·40-s − 0.312·41-s − 0.304·43-s − 0.150·44-s − 0.884·46-s + 0.291·47-s − 49-s − 0.141·50-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.46266480430679, −12.72296010737360, −12.64165941379814, −11.99361120395195, −11.36444371458911, −11.16383594625677, −10.62886597984426, −10.09261237460655, −9.691666457833211, −9.128990830147583, −8.648728446772395, −8.212418034494943, −7.697635877908338, −7.347197591673395, −6.626665459202390, −6.394921778447466, −5.612850762768487, −5.076397882728989, −4.586480839315145, −3.874544778000316, −3.282394480532675, −2.767216847439520, −2.171176472155364, −1.350237396935911, −0.7950458142659032, 0,
0.7950458142659032, 1.350237396935911, 2.171176472155364, 2.767216847439520, 3.282394480532675, 3.874544778000316, 4.586480839315145, 5.076397882728989, 5.612850762768487, 6.394921778447466, 6.626665459202390, 7.347197591673395, 7.697635877908338, 8.212418034494943, 8.648728446772395, 9.128990830147583, 9.691666457833211, 10.09261237460655, 10.62886597984426, 11.16383594625677, 11.36444371458911, 11.99361120395195, 12.64165941379814, 12.72296010737360, 13.46266480430679
