| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 4·7-s − 3·8-s + 9-s + 10-s − 4·11-s − 12-s + 6·13-s − 4·14-s + 15-s − 16-s + 2·17-s + 18-s + 4·19-s − 20-s − 4·21-s − 4·22-s − 3·24-s + 25-s + 6·26-s + 27-s + 4·28-s − 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.66·13-s − 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.872·21-s − 0.852·22-s − 0.612·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.755·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.266763313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266763313\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 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 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−7.71883373629564508091001212349, −7.21117919215508187731156381414, −6.00420600526633725738658493847, −5.89720106635173540153814128323, −5.15917295763919256192923395563, −4.05294501428975910255680366022, −3.36960191194525986630871818599, −3.15552036995145299536809538741, −2.02050709206461068285279914650, −0.63481371030913013007614086911,
0.63481371030913013007614086911, 2.02050709206461068285279914650, 3.15552036995145299536809538741, 3.36960191194525986630871818599, 4.05294501428975910255680366022, 5.15917295763919256192923395563, 5.89720106635173540153814128323, 6.00420600526633725738658493847, 7.21117919215508187731156381414, 7.71883373629564508091001212349
