| L(s) = 1 | + 5-s + 7-s − 3·9-s − 4·11-s − 2·13-s − 2·17-s + 19-s + 25-s − 10·29-s − 8·31-s + 35-s + 6·37-s + 10·41-s − 8·43-s − 3·45-s + 49-s − 10·53-s − 4·55-s − 4·59-s − 2·61-s − 3·63-s − 2·65-s + 12·67-s + 4·71-s − 2·73-s − 4·77-s + 12·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.229·19-s + 1/5·25-s − 1.85·29-s − 1.43·31-s + 0.169·35-s + 0.986·37-s + 1.56·41-s − 1.21·43-s − 0.447·45-s + 1/7·49-s − 1.37·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.377·63-s − 0.248·65-s + 1.46·67-s + 0.474·71-s − 0.234·73-s − 0.455·77-s + 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7778923485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7778923485\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−14.68251533564084, −14.30494897350439, −13.63599799969033, −13.15845633931375, −12.73440257658048, −12.21701260874829, −11.34235362664891, −11.04673623938668, −10.79176897958550, −9.839587042684701, −9.469128717909097, −8.987145582262266, −8.254607013751603, −7.756753641527458, −7.392500888425731, −6.551891403066440, −5.919752576852946, −5.303672734441798, −5.141512588508478, −4.228191023598682, −3.484095327544717, −2.707355230018315, −2.281292613080752, −1.546536828711620, −0.2997505146936820,
0.2997505146936820, 1.546536828711620, 2.281292613080752, 2.707355230018315, 3.484095327544717, 4.228191023598682, 5.141512588508478, 5.303672734441798, 5.919752576852946, 6.551891403066440, 7.392500888425731, 7.756753641527458, 8.254607013751603, 8.987145582262266, 9.469128717909097, 9.839587042684701, 10.79176897958550, 11.04673623938668, 11.34235362664891, 12.21701260874829, 12.73440257658048, 13.15845633931375, 13.63599799969033, 14.30494897350439, 14.68251533564084
