| L(s) = 1 | − 3·5-s − 2·7-s − 11-s + 6·17-s − 4·19-s − 23-s + 4·25-s − 8·29-s − 7·31-s + 6·35-s + 37-s − 4·41-s − 6·43-s + 8·47-s − 3·49-s + 2·53-s + 3·55-s − 59-s − 4·61-s + 5·67-s − 3·71-s + 16·73-s + 2·77-s + 2·79-s − 2·83-s − 18·85-s − 15·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s − 0.301·11-s + 1.45·17-s − 0.917·19-s − 0.208·23-s + 4/5·25-s − 1.48·29-s − 1.25·31-s + 1.01·35-s + 0.164·37-s − 0.624·41-s − 0.914·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.404·55-s − 0.130·59-s − 0.512·61-s + 0.610·67-s − 0.356·71-s + 1.87·73-s + 0.227·77-s + 0.225·79-s − 0.219·83-s − 1.95·85-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6758238763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6758238763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−7.990691719045278208721222124135, −7.37849424424332763125175440867, −6.81172608748271012882089204159, −5.84528783154170478811168793636, −5.24353420321932260945912450328, −4.16390592494301360488588998352, −3.65007101647416280205748713613, −3.03422982849807009100475551361, −1.82093737157603507455602198547, −0.41365712755271202009732742448,
0.41365712755271202009732742448, 1.82093737157603507455602198547, 3.03422982849807009100475551361, 3.65007101647416280205748713613, 4.16390592494301360488588998352, 5.24353420321932260945912450328, 5.84528783154170478811168793636, 6.81172608748271012882089204159, 7.37849424424332763125175440867, 7.990691719045278208721222124135
