| L(s) = 1 | + 3·3-s + 5-s + 3·7-s + 6·9-s + 4·13-s + 3·15-s − 4·19-s + 9·21-s + 8·23-s + 25-s + 9·27-s + 6·29-s + 2·31-s + 3·35-s − 8·37-s + 12·39-s − 5·41-s − 5·43-s + 6·45-s + 3·47-s + 2·49-s + 4·53-s − 12·57-s + 2·59-s − 11·61-s + 18·63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 1.13·7-s + 2·9-s + 1.10·13-s + 0.774·15-s − 0.917·19-s + 1.96·21-s + 1.66·23-s + 1/5·25-s + 1.73·27-s + 1.11·29-s + 0.359·31-s + 0.507·35-s − 1.31·37-s + 1.92·39-s − 0.780·41-s − 0.762·43-s + 0.894·45-s + 0.437·47-s + 2/7·49-s + 0.549·53-s − 1.58·57-s + 0.260·59-s − 1.40·61-s + 2.26·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.049654768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.049654768\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.985826184415954184985525437129, −7.02704525205786809216852805280, −6.63449952634640616684822458534, −5.50555993589618781107129374913, −4.74572303928508501883430909257, −4.11039932928135609359317489243, −3.26469072205859094381584102917, −2.65566738201317002581219526204, −1.74676347167216194287441290537, −1.24689197411205407079746822852,
1.24689197411205407079746822852, 1.74676347167216194287441290537, 2.65566738201317002581219526204, 3.26469072205859094381584102917, 4.11039932928135609359317489243, 4.74572303928508501883430909257, 5.50555993589618781107129374913, 6.63449952634640616684822458534, 7.02704525205786809216852805280, 7.985826184415954184985525437129
