| L(s) = 1 | − 6·13-s + 4·17-s + 4·19-s − 6·23-s − 5·25-s + 8·29-s − 4·31-s + 37-s − 10·41-s − 8·43-s + 8·47-s − 7·49-s + 6·53-s − 2·59-s − 10·61-s − 12·67-s + 8·71-s − 10·73-s + 8·79-s − 8·83-s − 16·89-s + 6·97-s − 6·101-s + 16·103-s + 12·107-s − 14·109-s + ⋯ |
| L(s) = 1 | − 1.66·13-s + 0.970·17-s + 0.917·19-s − 1.25·23-s − 25-s + 1.48·29-s − 0.718·31-s + 0.164·37-s − 1.56·41-s − 1.21·43-s + 1.16·47-s − 49-s + 0.824·53-s − 0.260·59-s − 1.28·61-s − 1.46·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s − 0.878·83-s − 1.69·89-s + 0.609·97-s − 0.597·101-s + 1.57·103-s + 1.16·107-s − 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.349778374652070522944483171008, −7.67419897828900331555845121123, −7.11580588360178380531607038280, −6.10393084738709341744128035876, −5.31488880682773694348775332571, −4.61597339575994533252765120951, −3.55300849819846767635896768081, −2.66952151423569177966024936577, −1.57898852442273499679951831437, 0,
1.57898852442273499679951831437, 2.66952151423569177966024936577, 3.55300849819846767635896768081, 4.61597339575994533252765120951, 5.31488880682773694348775332571, 6.10393084738709341744128035876, 7.11580588360178380531607038280, 7.67419897828900331555845121123, 8.349778374652070522944483171008
