| L(s) = 1 | − 5-s − 4·7-s − 3·9-s − 4·13-s + 7·17-s − 4·19-s − 2·23-s − 4·25-s − 4·29-s + 6·31-s + 4·35-s + 37-s + 2·41-s + 2·43-s + 3·45-s + 11·47-s + 9·49-s − 4·59-s − 10·61-s + 12·63-s + 4·65-s − 8·67-s − 3·71-s − 12·73-s − 13·79-s + 9·81-s + 83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 9-s − 1.10·13-s + 1.69·17-s − 0.917·19-s − 0.417·23-s − 4/5·25-s − 0.742·29-s + 1.07·31-s + 0.676·35-s + 0.164·37-s + 0.312·41-s + 0.304·43-s + 0.447·45-s + 1.60·47-s + 9/7·49-s − 0.520·59-s − 1.28·61-s + 1.51·63-s + 0.496·65-s − 0.977·67-s − 0.356·71-s − 1.40·73-s − 1.46·79-s + 81-s + 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 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
−14.66090916487413, −14.09114171609361, −13.67091887948462, −13.06650132731245, −12.45720029088483, −12.13050682640215, −11.87262628751935, −11.11683571996813, −10.44980434144649, −10.08366454633320, −9.550051755633895, −9.170714905072046, −8.470358262901155, −7.919239221600149, −7.424758877691220, −6.976749005243319, −6.064633460415265, −5.919980337981418, −5.384291406944946, −4.384874196249140, −4.030046433450839, −3.158373618277965, −2.914659693022912, −2.265102799873494, −1.178810052254221, 0, 0,
1.178810052254221, 2.265102799873494, 2.914659693022912, 3.158373618277965, 4.030046433450839, 4.384874196249140, 5.384291406944946, 5.919980337981418, 6.064633460415265, 6.976749005243319, 7.424758877691220, 7.919239221600149, 8.470358262901155, 9.170714905072046, 9.550051755633895, 10.08366454633320, 10.44980434144649, 11.11683571996813, 11.87262628751935, 12.13050682640215, 12.45720029088483, 13.06650132731245, 13.67091887948462, 14.09114171609361, 14.66090916487413
