| L(s) = 1 | − 2·5-s + 4·7-s + 6·17-s − 4·23-s − 25-s + 29-s − 6·31-s − 8·35-s − 4·37-s − 6·41-s + 4·43-s + 9·49-s − 2·53-s − 2·59-s − 12·67-s − 14·73-s + 10·79-s + 6·83-s − 12·85-s + 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·115-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.185·29-s − 1.07·31-s − 1.35·35-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.274·53-s − 0.260·59-s − 1.46·67-s − 1.63·73-s + 1.12·79-s + 0.658·83-s − 1.30·85-s + 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.746·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.62444760052487, −12.09916706521321, −11.82721377618492, −11.61119130421844, −10.94442948163696, −10.60589747073303, −10.15956660879067, −9.634422218425271, −8.935891142502701, −8.643358052715223, −7.979969751005268, −7.803474767995195, −7.453521181496477, −6.954905122383141, −6.123364012947076, −5.742570318679985, −5.177047301281108, −4.784583490677446, −4.249523502724755, −3.706216898881324, −3.355443547205747, −2.615673194875587, −1.786091666871226, −1.570394514983075, −0.7634477930942337, 0,
0.7634477930942337, 1.570394514983075, 1.786091666871226, 2.615673194875587, 3.355443547205747, 3.706216898881324, 4.249523502724755, 4.784583490677446, 5.177047301281108, 5.742570318679985, 6.123364012947076, 6.954905122383141, 7.453521181496477, 7.803474767995195, 7.979969751005268, 8.643358052715223, 8.935891142502701, 9.634422218425271, 10.15956660879067, 10.60589747073303, 10.94442948163696, 11.61119130421844, 11.82721377618492, 12.09916706521321, 12.62444760052487
