| L(s) = 1 | + 3-s − 5-s + 3·7-s − 2·9-s − 11-s − 13-s − 15-s + 3·17-s + 3·21-s + 4·23-s − 4·25-s − 5·27-s − 8·31-s − 33-s − 3·35-s + 7·37-s − 39-s − 8·41-s + 43-s + 2·45-s − 7·47-s + 2·49-s + 3·51-s + 6·53-s + 55-s − 10·59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.727·17-s + 0.654·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s − 1.43·31-s − 0.174·33-s − 0.507·35-s + 1.15·37-s − 0.160·39-s − 1.24·41-s + 0.152·43-s + 0.298·45-s − 1.02·47-s + 2/7·49-s + 0.420·51-s + 0.824·53-s + 0.134·55-s − 1.30·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−7.68841018431844673282410160755, −6.90698938499111254450545513162, −5.83475650794534987321119229155, −5.29710518067052618995715239101, −4.61445084287403233742781467464, −3.73853169386082124622793791060, −3.07613455484015371329987635123, −2.22228431697090480707132888876, −1.37464607858279854418184335965, 0,
1.37464607858279854418184335965, 2.22228431697090480707132888876, 3.07613455484015371329987635123, 3.73853169386082124622793791060, 4.61445084287403233742781467464, 5.29710518067052618995715239101, 5.83475650794534987321119229155, 6.90698938499111254450545513162, 7.68841018431844673282410160755
