| L(s) = 1 | − 3-s − 2·5-s + 2·7-s − 2·9-s + 4·11-s + 4·13-s + 2·15-s − 7·17-s + 3·19-s − 2·21-s − 25-s + 5·27-s − 4·29-s + 6·31-s − 4·33-s − 4·35-s + 2·37-s − 4·39-s + 6·41-s + 5·43-s + 4·45-s − 10·47-s − 3·49-s + 7·51-s − 8·55-s − 3·57-s − 5·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s − 2/3·9-s + 1.20·11-s + 1.10·13-s + 0.516·15-s − 1.69·17-s + 0.688·19-s − 0.436·21-s − 1/5·25-s + 0.962·27-s − 0.742·29-s + 1.07·31-s − 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.640·39-s + 0.937·41-s + 0.762·43-s + 0.596·45-s − 1.45·47-s − 3/7·49-s + 0.980·51-s − 1.07·55-s − 0.397·57-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.291418799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291418799\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.351785870380230845146811791906, −7.80948930884184229613886707010, −6.75936022229572744591890526054, −6.30000674831340909127108649748, −5.46368215983371937142564286862, −4.49125078122561128359473406677, −4.03506203907463739292817371199, −3.06293690750401355230592469619, −1.77662906913622074514743874887, −0.67881973729472760646099230435,
0.67881973729472760646099230435, 1.77662906913622074514743874887, 3.06293690750401355230592469619, 4.03506203907463739292817371199, 4.49125078122561128359473406677, 5.46368215983371937142564286862, 6.30000674831340909127108649748, 6.75936022229572744591890526054, 7.80948930884184229613886707010, 8.351785870380230845146811791906
