| L(s) = 1 | − 2·3-s + 4·5-s − 7-s + 9-s − 8·15-s + 2·17-s − 2·19-s + 2·21-s + 8·23-s + 11·25-s + 4·27-s + 2·29-s + 4·31-s − 4·35-s + 6·37-s + 2·41-s + 8·43-s + 4·45-s − 4·47-s + 49-s − 4·51-s + 10·53-s + 4·57-s − 6·59-s + 4·61-s − 63-s + 12·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 2.06·15-s + 0.485·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s + 11/5·25-s + 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.676·35-s + 0.986·37-s + 0.312·41-s + 1.21·43-s + 0.596·45-s − 0.583·47-s + 1/7·49-s − 0.560·51-s + 1.37·53-s + 0.529·57-s − 0.781·59-s + 0.512·61-s − 0.125·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.876685026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.876685026\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.41092887110111, −13.80361544943950, −13.37362690088019, −12.87082501222965, −12.45839935430298, −11.97392366696250, −11.10796030881363, −10.89967725459783, −10.37972939428556, −9.778283388692003, −9.459127466717601, −8.861326320914374, −8.282646428418105, −7.375439534296476, −6.750341391187713, −6.286721380659779, −6.037601900328191, −5.224233472141577, −5.134841039190967, −4.342580632425200, −3.345646256004185, −2.622714313080724, −2.174634431080021, −1.047461229532564, −0.7842173292074421,
0.7842173292074421, 1.047461229532564, 2.174634431080021, 2.622714313080724, 3.345646256004185, 4.342580632425200, 5.134841039190967, 5.224233472141577, 6.037601900328191, 6.286721380659779, 6.750341391187713, 7.375439534296476, 8.282646428418105, 8.861326320914374, 9.459127466717601, 9.778283388692003, 10.37972939428556, 10.89967725459783, 11.10796030881363, 11.97392366696250, 12.45839935430298, 12.87082501222965, 13.37362690088019, 13.80361544943950, 14.41092887110111
