| L(s) = 1 | + 3-s + 4·5-s + 9-s − 2·11-s + 13-s + 4·15-s + 2·17-s + 4·23-s + 11·25-s + 27-s + 6·29-s + 4·31-s − 2·33-s − 6·37-s + 39-s + 12·41-s + 4·43-s + 4·45-s − 6·47-s − 7·49-s + 2·51-s + 2·53-s − 8·55-s + 14·59-s + 10·61-s + 4·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 1.03·15-s + 0.485·17-s + 0.834·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.160·39-s + 1.87·41-s + 0.609·43-s + 0.596·45-s − 0.875·47-s − 49-s + 0.280·51-s + 0.274·53-s − 1.07·55-s + 1.82·59-s + 1.28·61-s + 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.531975995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.531975995\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−13.63726254287755, −13.25134206963356, −12.75109846246983, −12.49830021741645, −11.66607090354074, −11.06294587875023, −10.52386087503674, −10.12151814003598, −9.752702293447590, −9.283336642145954, −8.701457439824945, −8.376219074427985, −7.682485118693067, −7.112377107097969, −6.438446316438935, −6.223517008734533, −5.399292472259798, −5.133920484596847, −4.543836033264695, −3.658475537147040, −3.072852877583353, −2.418367986765562, −2.193865481573286, −1.243603091440923, −0.8255009435790962,
0.8255009435790962, 1.243603091440923, 2.193865481573286, 2.418367986765562, 3.072852877583353, 3.658475537147040, 4.543836033264695, 5.133920484596847, 5.399292472259798, 6.223517008734533, 6.438446316438935, 7.112377107097969, 7.682485118693067, 8.376219074427985, 8.701457439824945, 9.283336642145954, 9.752702293447590, 10.12151814003598, 10.52386087503674, 11.06294587875023, 11.66607090354074, 12.49830021741645, 12.75109846246983, 13.25134206963356, 13.63726254287755
