| L(s) = 1 | − 2·3-s + 9-s − 11-s + 3·13-s − 4·17-s + 19-s − 3·23-s + 4·27-s + 5·29-s + 3·31-s + 2·33-s − 12·37-s − 6·39-s + 8·41-s + 5·43-s + 8·47-s − 7·49-s + 8·51-s − 10·53-s − 2·57-s − 8·59-s + 10·61-s − 14·67-s + 6·69-s + 5·71-s − 4·73-s + 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.970·17-s + 0.229·19-s − 0.625·23-s + 0.769·27-s + 0.928·29-s + 0.538·31-s + 0.348·33-s − 1.97·37-s − 0.960·39-s + 1.24·41-s + 0.762·43-s + 1.16·47-s − 49-s + 1.12·51-s − 1.37·53-s − 0.264·57-s − 1.04·59-s + 1.28·61-s − 1.71·67-s + 0.722·69-s + 0.593·71-s − 0.468·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−8.009396201086558330755980077068, −7.09661170954696028589073275698, −6.31774208635278566143098758450, −5.95251555269509447771927127170, −5.04137137530561945884073121971, −4.44889853685944012773215613315, −3.44307601477960793257883007200, −2.37942017148615794672168448724, −1.15845751029152603421619719945, 0,
1.15845751029152603421619719945, 2.37942017148615794672168448724, 3.44307601477960793257883007200, 4.44889853685944012773215613315, 5.04137137530561945884073121971, 5.95251555269509447771927127170, 6.31774208635278566143098758450, 7.09661170954696028589073275698, 8.009396201086558330755980077068
