| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 15-s + 2·19-s − 2·21-s + 6·23-s + 25-s − 27-s + 8·31-s − 2·35-s − 2·37-s − 6·41-s + 4·43-s − 45-s − 3·49-s − 6·53-s − 2·57-s + 14·61-s + 2·63-s − 4·67-s − 6·69-s + 4·73-s − 75-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.258·15-s + 0.458·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 3/7·49-s − 0.824·53-s − 0.264·57-s + 1.79·61-s + 0.251·63-s − 0.488·67-s − 0.722·69-s + 0.468·73-s − 0.115·75-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.058985790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058985790\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−14.95377341727819, −14.14089480648404, −13.87830659257339, −13.07298243428581, −12.69273700285350, −12.03828069203349, −11.54698133039235, −11.27178201088711, −10.66621755314729, −10.12129124291499, −9.559996925598421, −8.831806071297857, −8.356090337809424, −7.778530310007246, −7.237980040925271, −6.661087382091659, −6.118420915472901, −5.245492959251591, −4.945274766689368, −4.391207889672392, −3.591182250988087, −2.969773710823163, −2.094117245068243, −1.240954886467758, −0.6070440577406721,
0.6070440577406721, 1.240954886467758, 2.094117245068243, 2.969773710823163, 3.591182250988087, 4.391207889672392, 4.945274766689368, 5.245492959251591, 6.118420915472901, 6.661087382091659, 7.237980040925271, 7.778530310007246, 8.356090337809424, 8.831806071297857, 9.559996925598421, 10.12129124291499, 10.66621755314729, 11.27178201088711, 11.54698133039235, 12.03828069203349, 12.69273700285350, 13.07298243428581, 13.87830659257339, 14.14089480648404, 14.95377341727819
