| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s + 13-s − 2·15-s − 2·17-s − 21-s − 3·23-s − 25-s − 27-s + 29-s − 7·31-s + 2·35-s + 4·37-s − 39-s + 5·41-s − 43-s + 2·45-s + 2·47-s + 49-s + 2·51-s + 10·53-s − 3·59-s + 61-s + 63-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s − 0.485·17-s − 0.218·21-s − 0.625·23-s − 1/5·25-s − 0.192·27-s + 0.185·29-s − 1.25·31-s + 0.338·35-s + 0.657·37-s − 0.160·39-s + 0.780·41-s − 0.152·43-s + 0.298·45-s + 0.291·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.390·59-s + 0.128·61-s + 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112360234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112360234\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.72852005564108, −14.23044846735549, −13.64504963227157, −13.25375684525629, −12.74611091612556, −12.14087289157279, −11.56296350107598, −11.15555226468157, −10.45635661823008, −10.23733718440506, −9.374662391464766, −9.151596153412345, −8.383597845078089, −7.743293610009626, −7.201924755509977, −6.498714747790705, −6.002226750417577, −5.548566897311193, −4.997963943343787, −4.221095193238933, −3.763515612983067, −2.709751880214748, −2.069404735848108, −1.478806362115812, −0.5504777759138855,
0.5504777759138855, 1.478806362115812, 2.069404735848108, 2.709751880214748, 3.763515612983067, 4.221095193238933, 4.997963943343787, 5.548566897311193, 6.002226750417577, 6.498714747790705, 7.201924755509977, 7.743293610009626, 8.383597845078089, 9.151596153412345, 9.374662391464766, 10.23733718440506, 10.45635661823008, 11.15555226468157, 11.56296350107598, 12.14087289157279, 12.74611091612556, 13.25375684525629, 13.64504963227157, 14.23044846735549, 14.72852005564108
