| L(s) = 1 | − 2-s − 4-s + 3·8-s − 2·11-s − 16-s + 2·19-s + 2·22-s − 8·23-s + 10·29-s + 4·31-s − 5·32-s − 2·37-s − 2·38-s + 41-s + 4·43-s + 2·44-s + 8·46-s − 12·47-s − 7·49-s − 10·58-s + 2·61-s − 4·62-s + 7·64-s + 8·67-s − 14·71-s + 6·73-s + 2·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 0.603·11-s − 1/4·16-s + 0.458·19-s + 0.426·22-s − 1.66·23-s + 1.85·29-s + 0.718·31-s − 0.883·32-s − 0.328·37-s − 0.324·38-s + 0.156·41-s + 0.609·43-s + 0.301·44-s + 1.17·46-s − 1.75·47-s − 49-s − 1.31·58-s + 0.256·61-s − 0.508·62-s + 7/8·64-s + 0.977·67-s − 1.66·71-s + 0.702·73-s + 0.232·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−7.75656259461067281277387926660, −6.74578199961958040654940166296, −6.12150972563998659950327970177, −5.16869321327189781063020525960, −4.67588143571028800147872419794, −3.88907936104043427389408274653, −2.98351381479084191222461142702, −2.01802149788384614020875370689, −1.03195760180061048985914304025, 0,
1.03195760180061048985914304025, 2.01802149788384614020875370689, 2.98351381479084191222461142702, 3.88907936104043427389408274653, 4.67588143571028800147872419794, 5.16869321327189781063020525960, 6.12150972563998659950327970177, 6.74578199961958040654940166296, 7.75656259461067281277387926660
