| L(s) = 1 | − 3-s + 4·5-s − 4·7-s + 9-s + 4·11-s − 6·13-s − 4·15-s + 2·17-s − 19-s + 4·21-s + 4·23-s + 11·25-s − 27-s + 10·29-s − 4·31-s − 4·33-s − 16·35-s − 6·37-s + 6·39-s + 8·41-s − 43-s + 4·45-s + 4·47-s + 9·49-s − 2·51-s + 12·53-s + 16·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 1.03·15-s + 0.485·17-s − 0.229·19-s + 0.872·21-s + 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.696·33-s − 2.70·35-s − 0.986·37-s + 0.960·39-s + 1.24·41-s − 0.152·43-s + 0.596·45-s + 0.583·47-s + 9/7·49-s − 0.280·51-s + 1.64·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.388658078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.388658078\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.66575909099882, −14.28448663298660, −13.69281789075499, −13.28875510264563, −12.61647239810256, −12.25853850115684, −12.01480680321189, −10.92949343419898, −10.45269257739304, −9.823213831616474, −9.806975508886919, −9.013858477551133, −8.880730708028250, −7.560469636947694, −6.947137486472600, −6.587504122131622, −6.187808899149394, −5.534732614985075, −5.078626759367548, −4.364840117237791, −3.484169075310657, −2.738039791067075, −2.289638870955912, −1.338339193530047, −0.6137296281887823,
0.6137296281887823, 1.338339193530047, 2.289638870955912, 2.738039791067075, 3.484169075310657, 4.364840117237791, 5.078626759367548, 5.534732614985075, 6.187808899149394, 6.587504122131622, 6.947137486472600, 7.560469636947694, 8.880730708028250, 9.013858477551133, 9.806975508886919, 9.823213831616474, 10.45269257739304, 10.92949343419898, 12.01480680321189, 12.25853850115684, 12.61647239810256, 13.28875510264563, 13.69281789075499, 14.28448663298660, 14.66575909099882
