| L(s) = 1 | − 5-s − 11-s − 7·13-s + 2·17-s + 2·23-s + 25-s − 2·29-s − 3·31-s + 12·37-s + 6·41-s − 43-s + 10·47-s + 55-s − 7·59-s + 7·65-s + 4·67-s + 9·71-s − 9·73-s + 6·79-s − 11·83-s − 2·85-s + 7·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.94·13-s + 0.485·17-s + 0.417·23-s + 1/5·25-s − 0.371·29-s − 0.538·31-s + 1.97·37-s + 0.937·41-s − 0.152·43-s + 1.45·47-s + 0.134·55-s − 0.911·59-s + 0.868·65-s + 0.488·67-s + 1.06·71-s − 1.05·73-s + 0.675·79-s − 1.20·83-s − 0.216·85-s + 0.741·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330851320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330851320\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.46450425716737, −12.06110070980609, −11.55051909565828, −11.18143494081272, −10.67048030742855, −10.14956277077480, −9.788857968899782, −9.252025761513610, −9.011102332747595, −8.236853419723608, −7.750519159679654, −7.458919404831716, −7.172083883078133, −6.514805546156044, −5.818274062646116, −5.566990163948323, −4.844113306061545, −4.528473085792094, −4.088710068512011, −3.254083115236705, −2.951904057535175, −2.261238264999069, −1.921914785439941, −0.7953127369186482, −0.5217526184498437,
0.5217526184498437, 0.7953127369186482, 1.921914785439941, 2.261238264999069, 2.951904057535175, 3.254083115236705, 4.088710068512011, 4.528473085792094, 4.844113306061545, 5.566990163948323, 5.818274062646116, 6.514805546156044, 7.172083883078133, 7.458919404831716, 7.750519159679654, 8.236853419723608, 9.011102332747595, 9.252025761513610, 9.788857968899782, 10.14956277077480, 10.67048030742855, 11.18143494081272, 11.55051909565828, 12.06110070980609, 12.46450425716737
