| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 6·13-s − 14-s + 16-s − 2·17-s − 2·19-s + 20-s + 2·23-s − 4·25-s − 6·26-s + 28-s + 6·29-s − 32-s + 2·34-s + 35-s + 3·37-s + 2·38-s − 40-s + 9·41-s − 8·43-s − 2·46-s + 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.458·19-s + 0.223·20-s + 0.417·23-s − 4/5·25-s − 1.17·26-s + 0.188·28-s + 1.11·29-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.493·37-s + 0.324·38-s − 0.158·40-s + 1.40·41-s − 1.21·43-s − 0.294·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.445561569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.445561569\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 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
−14.67648663185318, −14.03300348080820, −13.66495354173781, −13.04789022030158, −12.67422560355476, −11.83031877511619, −11.35651134126452, −11.01496630788182, −10.42533382949698, −9.960284375658030, −9.309445199907716, −8.818152023990265, −8.297154396862760, −7.989916513678061, −7.149531663070710, −6.473789220366379, −6.250116835429370, −5.490734744317966, −4.909815651395740, −3.965399730912059, −3.637372659938832, −2.540118314528374, −2.141428581462341, −1.237503984789615, −0.6983189729323091,
0.6983189729323091, 1.237503984789615, 2.141428581462341, 2.540118314528374, 3.637372659938832, 3.965399730912059, 4.909815651395740, 5.490734744317966, 6.250116835429370, 6.473789220366379, 7.149531663070710, 7.989916513678061, 8.297154396862760, 8.818152023990265, 9.309445199907716, 9.960284375658030, 10.42533382949698, 11.01496630788182, 11.35651134126452, 11.83031877511619, 12.67422560355476, 13.04789022030158, 13.66495354173781, 14.03300348080820, 14.67648663185318
