| L(s) = 1 | − 2·2-s + 2·4-s − 7-s − 6·11-s − 7·13-s + 2·14-s − 4·16-s − 4·17-s + 5·19-s + 12·22-s − 2·23-s + 14·26-s − 2·28-s − 8·29-s + 3·31-s + 8·32-s + 8·34-s + 37-s − 10·38-s + 2·41-s − 11·43-s − 12·44-s + 4·46-s − 4·47-s − 6·49-s − 14·52-s + 16·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.377·7-s − 1.80·11-s − 1.94·13-s + 0.534·14-s − 16-s − 0.970·17-s + 1.14·19-s + 2.55·22-s − 0.417·23-s + 2.74·26-s − 0.377·28-s − 1.48·29-s + 0.538·31-s + 1.41·32-s + 1.37·34-s + 0.164·37-s − 1.62·38-s + 0.312·41-s − 1.67·43-s − 1.80·44-s + 0.589·46-s − 0.583·47-s − 6/7·49-s − 1.94·52-s + 2.10·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−7.27229501326324136892848590776, −6.91414296710153130903718957926, −5.75143573944665722335908266690, −5.03323063277497513632701768766, −4.46422935061264766509686941457, −3.05812908162585319931345484493, −2.48339441959668457066389074274, −1.66363019418424139295817124892, 0, 0,
1.66363019418424139295817124892, 2.48339441959668457066389074274, 3.05812908162585319931345484493, 4.46422935061264766509686941457, 5.03323063277497513632701768766, 5.75143573944665722335908266690, 6.91414296710153130903718957926, 7.27229501326324136892848590776
