| L(s) = 1 | + 3-s + 4·5-s − 7-s + 9-s + 4·15-s + 2·17-s − 21-s + 4·23-s + 11·25-s + 27-s − 2·29-s + 8·31-s − 4·35-s + 8·37-s + 4·41-s + 8·43-s + 4·45-s + 49-s + 2·51-s − 6·53-s + 8·59-s + 10·61-s − 63-s − 8·67-s + 4·69-s + 4·73-s + 11·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 1.03·15-s + 0.485·17-s − 0.218·21-s + 0.834·23-s + 11/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.676·35-s + 1.31·37-s + 0.624·41-s + 1.21·43-s + 0.596·45-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 1.04·59-s + 1.28·61-s − 0.125·63-s − 0.977·67-s + 0.481·69-s + 0.468·73-s + 1.27·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.176987436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.176987436\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.08244300704637, −12.67441011700698, −12.23321947404066, −11.36755600991151, −11.13247433667927, −10.29257904054999, −10.08956464229139, −9.747221179387642, −9.106627556416449, −8.990690086227647, −8.342286505329651, −7.681182813959291, −7.274756761877685, −6.616431872715452, −6.172778495049694, −5.871590295082811, −5.237122919783121, −4.743396775019197, −4.170148936614560, −3.392807709759053, −2.861782901513257, −2.430235299090441, −1.974943134227981, −1.119111659229729, −0.7975815282990905,
0.7975815282990905, 1.119111659229729, 1.974943134227981, 2.430235299090441, 2.861782901513257, 3.392807709759053, 4.170148936614560, 4.743396775019197, 5.237122919783121, 5.871590295082811, 6.172778495049694, 6.616431872715452, 7.274756761877685, 7.681182813959291, 8.342286505329651, 8.990690086227647, 9.106627556416449, 9.747221179387642, 10.08956464229139, 10.29257904054999, 11.13247433667927, 11.36755600991151, 12.23321947404066, 12.67441011700698, 13.08244300704637
