| L(s) = 1 | + 3-s + 5-s + 9-s + 5·11-s + 15-s + 4·17-s − 8·19-s + 4·23-s − 4·25-s + 27-s + 5·29-s + 3·31-s + 5·33-s + 4·37-s + 2·43-s + 45-s − 6·47-s + 4·51-s + 9·53-s + 5·55-s − 8·57-s + 11·59-s − 6·61-s − 2·67-s + 4·69-s − 2·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.50·11-s + 0.258·15-s + 0.970·17-s − 1.83·19-s + 0.834·23-s − 4/5·25-s + 0.192·27-s + 0.928·29-s + 0.538·31-s + 0.870·33-s + 0.657·37-s + 0.304·43-s + 0.149·45-s − 0.875·47-s + 0.560·51-s + 1.23·53-s + 0.674·55-s − 1.05·57-s + 1.43·59-s − 0.768·61-s − 0.244·67-s + 0.481·69-s − 0.237·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.419875652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.419875652\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−7.73421539395236366430001804493, −6.96541645915432855218597929462, −6.34679548015235741300450429702, −5.86016148269565957058685991509, −4.76714180035995561453715019228, −4.15734441221853704248790189745, −3.46995114737342128672316311526, −2.57126246579998573602374877697, −1.76638690175158197459294636409, −0.907169062729814406464088705719,
0.907169062729814406464088705719, 1.76638690175158197459294636409, 2.57126246579998573602374877697, 3.46995114737342128672316311526, 4.15734441221853704248790189745, 4.76714180035995561453715019228, 5.86016148269565957058685991509, 6.34679548015235741300450429702, 6.96541645915432855218597929462, 7.73421539395236366430001804493
