| L(s) = 1 | + 3-s + 5-s − 3·7-s − 2·9-s + 5·13-s + 15-s − 3·17-s + 19-s − 3·21-s − 7·23-s + 25-s − 5·27-s − 29-s − 2·31-s − 3·35-s + 2·37-s + 5·39-s − 10·41-s + 6·43-s − 2·45-s − 8·47-s + 2·49-s − 3·51-s + 9·53-s + 57-s − 5·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s + 1.38·13-s + 0.258·15-s − 0.727·17-s + 0.229·19-s − 0.654·21-s − 1.45·23-s + 1/5·25-s − 0.962·27-s − 0.185·29-s − 0.359·31-s − 0.507·35-s + 0.328·37-s + 0.800·39-s − 1.56·41-s + 0.914·43-s − 0.298·45-s − 1.16·47-s + 2/7·49-s − 0.420·51-s + 1.23·53-s + 0.132·57-s − 0.650·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.599899300331853808825966946556, −7.69468636171176330451454909391, −6.67673332670759410864233619588, −6.09992657672615849191625135976, −5.52252039255676236152375425944, −4.16370882343868699820105573880, −3.45350191462770659015005646222, −2.70046380547565883418315316541, −1.66313070048963364301501257760, 0,
1.66313070048963364301501257760, 2.70046380547565883418315316541, 3.45350191462770659015005646222, 4.16370882343868699820105573880, 5.52252039255676236152375425944, 6.09992657672615849191625135976, 6.67673332670759410864233619588, 7.69468636171176330451454909391, 8.599899300331853808825966946556
