| L(s) = 1 | + 2·5-s + 2·7-s − 3·9-s − 6·11-s − 13-s + 17-s − 19-s − 3·23-s − 25-s − 6·29-s − 9·31-s + 4·35-s − 37-s + 3·41-s − 3·43-s − 6·45-s − 6·47-s − 3·49-s + 6·53-s − 12·55-s + 5·59-s + 7·61-s − 6·63-s − 2·65-s + 3·67-s + 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 9-s − 1.80·11-s − 0.277·13-s + 0.242·17-s − 0.229·19-s − 0.625·23-s − 1/5·25-s − 1.11·29-s − 1.61·31-s + 0.676·35-s − 0.164·37-s + 0.468·41-s − 0.457·43-s − 0.894·45-s − 0.875·47-s − 3/7·49-s + 0.824·53-s − 1.61·55-s + 0.650·59-s + 0.896·61-s − 0.755·63-s − 0.248·65-s + 0.366·67-s + 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−8.714468199680868640199167376738, −7.992841858497340642119138750325, −7.42860116284048239333542391102, −6.18515243791468325744523930807, −5.34746621749888827445294896627, −5.17914235246828118884988101265, −3.70113458490097721825929356794, −2.52387993244914225818904652757, −1.91105420895903679879612966377, 0,
1.91105420895903679879612966377, 2.52387993244914225818904652757, 3.70113458490097721825929356794, 5.17914235246828118884988101265, 5.34746621749888827445294896627, 6.18515243791468325744523930807, 7.42860116284048239333542391102, 7.992841858497340642119138750325, 8.714468199680868640199167376738
