| L(s) = 1 | − 2·3-s − 5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 4·19-s − 4·23-s + 25-s + 4·27-s − 29-s − 4·31-s − 8·33-s − 8·37-s + 4·39-s − 2·41-s + 2·43-s − 45-s + 2·47-s − 7·49-s − 14·53-s − 4·55-s − 8·57-s + 4·59-s + 2·61-s + 2·65-s − 4·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.769·27-s − 0.185·29-s − 0.718·31-s − 1.39·33-s − 1.31·37-s + 0.640·39-s − 0.312·41-s + 0.304·43-s − 0.149·45-s + 0.291·47-s − 49-s − 1.92·53-s − 0.539·55-s − 1.05·57-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−9.474360837820346931524343134506, −8.613335518329501316977489966396, −7.53734756567832052771273550063, −6.79895271534160961493776491136, −5.98106107712783762622279997837, −5.16629921678639568458331282345, −4.26063959697257778611218178241, −3.21573380725867205109830072566, −1.50609836333013593904529574681, 0,
1.50609836333013593904529574681, 3.21573380725867205109830072566, 4.26063959697257778611218178241, 5.16629921678639568458331282345, 5.98106107712783762622279997837, 6.79895271534160961493776491136, 7.53734756567832052771273550063, 8.613335518329501316977489966396, 9.474360837820346931524343134506
