| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 15-s + 2·17-s + 5·19-s − 4·21-s + 23-s − 4·25-s + 27-s − 2·31-s + 4·33-s + 4·35-s + 4·37-s + 2·41-s − 43-s − 45-s + 7·47-s + 9·49-s + 2·51-s + 5·53-s − 4·55-s + 5·57-s − 59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.258·15-s + 0.485·17-s + 1.14·19-s − 0.872·21-s + 0.208·23-s − 4/5·25-s + 0.192·27-s − 0.359·31-s + 0.696·33-s + 0.676·35-s + 0.657·37-s + 0.312·41-s − 0.152·43-s − 0.149·45-s + 1.02·47-s + 9/7·49-s + 0.280·51-s + 0.686·53-s − 0.539·55-s + 0.662·57-s − 0.130·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.827807449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.827807449\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 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
−13.03127389582353, −12.77564223599129, −12.15514014803014, −11.76607508049362, −11.47029259513015, −10.65437293309315, −10.16526362803204, −9.705689348296093, −9.284860529184674, −9.065393092589477, −8.410256768262764, −7.743542560907326, −7.389323947956379, −6.889627057401240, −6.441948766562422, −5.813756057437710, −5.483738821513125, −4.505672438684136, −4.068655855816221, −3.514601664462183, −3.246845033796995, −2.629288602905128, −1.882371870013625, −1.070213981340595, −0.5252103383863097,
0.5252103383863097, 1.070213981340595, 1.882371870013625, 2.629288602905128, 3.246845033796995, 3.514601664462183, 4.068655855816221, 4.505672438684136, 5.483738821513125, 5.813756057437710, 6.441948766562422, 6.889627057401240, 7.389323947956379, 7.743542560907326, 8.410256768262764, 9.065393092589477, 9.284860529184674, 9.705689348296093, 10.16526362803204, 10.65437293309315, 11.47029259513015, 11.76607508049362, 12.15514014803014, 12.77564223599129, 13.03127389582353
