| L(s) = 1 | − 3-s − 2·7-s − 2·9-s − 13-s − 6·17-s − 2·19-s + 2·21-s + 23-s − 5·25-s + 5·27-s − 3·29-s − 5·31-s + 8·37-s + 39-s + 3·41-s − 8·43-s − 9·47-s − 3·49-s + 6·51-s + 6·53-s + 2·57-s + 12·59-s + 14·61-s + 4·63-s − 8·67-s − 69-s + 15·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.277·13-s − 1.45·17-s − 0.458·19-s + 0.436·21-s + 0.208·23-s − 25-s + 0.962·27-s − 0.557·29-s − 0.898·31-s + 1.31·37-s + 0.160·39-s + 0.468·41-s − 1.21·43-s − 1.31·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 1.79·61-s + 0.503·63-s − 0.977·67-s − 0.120·69-s + 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−11.16792878112545616987260265830, −10.05292017083867367601940145437, −9.179275350265405931647106340341, −8.212075228103815323528954683241, −6.89620203002587049775804567261, −6.16371681572141052386909884198, −5.13331484784391238836917003859, −3.83402862781800864661702948677, −2.39273654521761793285798715610, 0,
2.39273654521761793285798715610, 3.83402862781800864661702948677, 5.13331484784391238836917003859, 6.16371681572141052386909884198, 6.89620203002587049775804567261, 8.212075228103815323528954683241, 9.179275350265405931647106340341, 10.05292017083867367601940145437, 11.16792878112545616987260265830
