| L(s) = 1 | − 5-s − 2·7-s − 3·9-s + 4·13-s + 4·17-s + 25-s + 6·29-s + 2·35-s − 2·37-s − 6·41-s + 2·43-s + 3·45-s − 3·49-s − 10·53-s − 12·59-s + 6·61-s + 6·63-s − 4·65-s + 12·67-s − 16·71-s − 4·73-s − 4·79-s + 9·81-s + 2·83-s − 4·85-s + 6·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 9-s + 1.10·13-s + 0.970·17-s + 1/5·25-s + 1.11·29-s + 0.338·35-s − 0.328·37-s − 0.937·41-s + 0.304·43-s + 0.447·45-s − 3/7·49-s − 1.37·53-s − 1.56·59-s + 0.768·61-s + 0.755·63-s − 0.496·65-s + 1.46·67-s − 1.89·71-s − 0.468·73-s − 0.450·79-s + 81-s + 0.219·83-s − 0.433·85-s + 0.635·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.324483875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.324483875\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.84863294175403502907095803870, −6.88345650860510459922238836452, −6.30336277527995968800606959251, −5.74741088295056651919606011737, −4.98104381995250999987643711767, −4.07532433744834858444870367932, −3.19325780560767377445560084372, −3.01638532533087116670330297173, −1.63198300925213934144992690252, −0.55241827494851289982710173354,
0.55241827494851289982710173354, 1.63198300925213934144992690252, 3.01638532533087116670330297173, 3.19325780560767377445560084372, 4.07532433744834858444870367932, 4.98104381995250999987643711767, 5.74741088295056651919606011737, 6.30336277527995968800606959251, 6.88345650860510459922238836452, 7.84863294175403502907095803870
