| L(s) = 1 | + 3-s + 3·7-s + 9-s + 11-s + 4·13-s − 17-s − 2·19-s + 3·21-s − 6·23-s − 5·25-s + 27-s + 9·29-s + 4·31-s + 33-s − 2·37-s + 4·39-s − 41-s + 6·43-s + 9·47-s + 2·49-s − 51-s − 13·53-s − 2·57-s − 15·59-s + 14·61-s + 3·63-s − 9·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.242·17-s − 0.458·19-s + 0.654·21-s − 1.25·23-s − 25-s + 0.192·27-s + 1.67·29-s + 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s − 0.156·41-s + 0.914·43-s + 1.31·47-s + 2/7·49-s − 0.140·51-s − 1.78·53-s − 0.264·57-s − 1.95·59-s + 1.79·61-s + 0.377·63-s − 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.924657004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.924657004\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.83780866934875, −14.24754832143748, −13.92586763567421, −13.61558474470274, −12.88114569360489, −12.12081539966781, −11.90712965127970, −11.15353765499838, −10.71323530162763, −10.18751490963650, −9.532121656947780, −8.871277158476154, −8.431725791500816, −7.966409846180920, −7.590383746285792, −6.622573339043701, −6.206952431639466, −5.602257894658368, −4.620565360419243, −4.360008317676802, −3.682170080985102, −2.920304657274407, −2.067033559313784, −1.601046290202745, −0.7300782024525559,
0.7300782024525559, 1.601046290202745, 2.067033559313784, 2.920304657274407, 3.682170080985102, 4.360008317676802, 4.620565360419243, 5.602257894658368, 6.206952431639466, 6.622573339043701, 7.590383746285792, 7.966409846180920, 8.431725791500816, 8.871277158476154, 9.532121656947780, 10.18751490963650, 10.71323530162763, 11.15353765499838, 11.90712965127970, 12.12081539966781, 12.88114569360489, 13.61558474470274, 13.92586763567421, 14.24754832143748, 14.83780866934875
