| L(s) = 1 | − 2·5-s − 4·7-s + 2·13-s + 2·17-s − 4·23-s − 25-s − 29-s − 6·31-s + 8·35-s − 4·37-s + 2·41-s − 4·43-s + 8·47-s + 9·49-s − 14·53-s − 6·59-s − 8·61-s − 4·65-s + 12·67-s + 16·71-s − 2·73-s + 6·79-s + 2·83-s − 4·85-s + 14·89-s − 8·91-s − 14·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 0.554·13-s + 0.485·17-s − 0.834·23-s − 1/5·25-s − 0.185·29-s − 1.07·31-s + 1.35·35-s − 0.657·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s − 1.92·53-s − 0.781·59-s − 1.02·61-s − 0.496·65-s + 1.46·67-s + 1.89·71-s − 0.234·73-s + 0.675·79-s + 0.219·83-s − 0.433·85-s + 1.48·89-s − 0.838·91-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8200508626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8200508626\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.302815177956909444494673508438, −7.70085637161607242276561348850, −6.95531634392195080398628392499, −6.23481184642361627827716563051, −5.61404148235702123127586932569, −4.48849015006953166000777911511, −3.51219621675131571260604400693, −3.37556090880625386854031493014, −1.99815517922978288240723862908, −0.49364605628437786580884547866,
0.49364605628437786580884547866, 1.99815517922978288240723862908, 3.37556090880625386854031493014, 3.51219621675131571260604400693, 4.48849015006953166000777911511, 5.61404148235702123127586932569, 6.23481184642361627827716563051, 6.95531634392195080398628392499, 7.70085637161607242276561348850, 8.302815177956909444494673508438
