| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s + 13-s − 15-s + 4·19-s − 21-s − 4·23-s − 4·25-s + 27-s − 10·29-s + 2·31-s + 33-s + 35-s + 5·37-s + 39-s − 12·41-s + 7·43-s − 45-s + 6·47-s + 49-s − 53-s − 55-s + 4·57-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s + 0.917·19-s − 0.218·21-s − 0.834·23-s − 4/5·25-s + 0.192·27-s − 1.85·29-s + 0.359·31-s + 0.174·33-s + 0.169·35-s + 0.821·37-s + 0.160·39-s − 1.87·41-s + 1.06·43-s − 0.149·45-s + 0.875·47-s + 1/7·49-s − 0.137·53-s − 0.134·55-s + 0.529·57-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.009749721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009749721\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−14.62275006658415, −13.93242966778368, −13.60316896188962, −13.13704406085120, −12.51779833162435, −11.86841114017425, −11.66960961245564, −10.92077756605666, −10.38582433591250, −9.674865039747972, −9.422640701868822, −8.828270184905979, −8.185447953068350, −7.636808885991206, −7.343230753201942, −6.579815838914044, −5.944000406004841, −5.473727078784092, −4.613083425288164, −3.936670623371556, −3.600399829931582, −2.941448925115027, −2.124902933562293, −1.481617272652022, −0.4813114648659565,
0.4813114648659565, 1.481617272652022, 2.124902933562293, 2.941448925115027, 3.600399829931582, 3.936670623371556, 4.613083425288164, 5.473727078784092, 5.944000406004841, 6.579815838914044, 7.343230753201942, 7.636808885991206, 8.185447953068350, 8.828270184905979, 9.422640701868822, 9.674865039747972, 10.38582433591250, 10.92077756605666, 11.66960961245564, 11.86841114017425, 12.51779833162435, 13.13704406085120, 13.60316896188962, 13.93242966778368, 14.62275006658415
