| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s + 2·13-s − 6·17-s + 4·19-s − 21-s + 27-s − 6·29-s + 8·31-s + 33-s − 10·37-s + 2·39-s − 6·41-s + 8·43-s + 49-s − 6·51-s + 6·53-s + 4·57-s − 12·59-s − 2·61-s − 63-s − 4·67-s + 12·71-s + 10·73-s − 77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.218·21-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s − 1.64·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.125·63-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.88192415996077, −12.20398194536337, −11.91093423734975, −11.34779056982064, −10.92116946271918, −10.43873835589559, −10.00557931601078, −9.393472661297443, −9.139773013828230, −8.641491641016900, −8.330334296563090, −7.586718924061411, −7.314487426121120, −6.644916781243743, −6.427274000061906, −5.781097889723319, −5.238353792149150, −4.636403752889918, −4.208646032160804, −3.531565428242231, −3.320047761850859, −2.570507392266422, −2.076269447915473, −1.488201416733640, −0.8014252154999612, 0,
0.8014252154999612, 1.488201416733640, 2.076269447915473, 2.570507392266422, 3.320047761850859, 3.531565428242231, 4.208646032160804, 4.636403752889918, 5.238353792149150, 5.781097889723319, 6.427274000061906, 6.644916781243743, 7.314487426121120, 7.586718924061411, 8.330334296563090, 8.641491641016900, 9.139773013828230, 9.393472661297443, 10.00557931601078, 10.43873835589559, 10.92116946271918, 11.34779056982064, 11.91093423734975, 12.20398194536337, 12.88192415996077
