| L(s) = 1 | + 4·5-s − 6·13-s + 8·17-s + 11·25-s + 4·29-s + 2·37-s + 8·41-s − 7·49-s − 4·53-s + 10·61-s − 24·65-s + 6·73-s + 32·85-s + 16·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.66·13-s + 1.94·17-s + 11/5·25-s + 0.742·29-s + 0.328·37-s + 1.24·41-s − 49-s − 0.549·53-s + 1.28·61-s − 2.97·65-s + 0.702·73-s + 3.47·85-s + 1.69·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.700055360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.700055360\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.13403192415212, −13.00074636596531, −12.38883832737158, −12.07414270717428, −11.47243698589815, −10.76587222069120, −10.23429108183383, −9.931655857706771, −9.638496486992490, −9.208270087990035, −8.577628694594711, −7.876053527463506, −7.494249450487183, −6.958952160309186, −6.223938067180347, −6.007900246988381, −5.342406747745025, −4.966710387793882, −4.568048354582277, −3.499300390137449, −3.052998910478873, −2.332560412883347, −2.077554398211708, −1.198783640181417, −0.6744676392209566,
0.6744676392209566, 1.198783640181417, 2.077554398211708, 2.332560412883347, 3.052998910478873, 3.499300390137449, 4.568048354582277, 4.966710387793882, 5.342406747745025, 6.007900246988381, 6.223938067180347, 6.958952160309186, 7.494249450487183, 7.876053527463506, 8.577628694594711, 9.208270087990035, 9.638496486992490, 9.931655857706771, 10.23429108183383, 10.76587222069120, 11.47243698589815, 12.07414270717428, 12.38883832737158, 13.00074636596531, 13.13403192415212
