| L(s) = 1 | − 5-s − 7-s − 11-s − 13-s + 17-s − 3·19-s + 6·23-s + 25-s + 29-s + 7·31-s + 35-s + 7·37-s − 6·41-s − 4·43-s + 6·47-s − 6·49-s + 5·53-s + 55-s − 14·59-s + 13·61-s + 65-s − 8·67-s − 5·71-s + 10·73-s + 77-s − 2·79-s − 14·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s − 0.277·13-s + 0.242·17-s − 0.688·19-s + 1.25·23-s + 1/5·25-s + 0.185·29-s + 1.25·31-s + 0.169·35-s + 1.15·37-s − 0.937·41-s − 0.609·43-s + 0.875·47-s − 6/7·49-s + 0.686·53-s + 0.134·55-s − 1.82·59-s + 1.66·61-s + 0.124·65-s − 0.977·67-s − 0.593·71-s + 1.17·73-s + 0.113·77-s − 0.225·79-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.93517665015018, −13.45165215699206, −12.84178556049644, −12.67351147871960, −11.99816803273836, −11.51330289910840, −11.12429844038056, −10.45556947511850, −10.10068521046238, −9.583913557100016, −8.959412507287587, −8.492755620147226, −8.009942880801348, −7.492166044330213, −6.815353993429146, −6.580975420045259, −5.815721586477977, −5.298024605638520, −4.572975431529405, −4.316510491518577, −3.452689413053958, −2.929508539458880, −2.483908247174668, −1.539712246092811, −0.8053379872680947, 0,
0.8053379872680947, 1.539712246092811, 2.483908247174668, 2.929508539458880, 3.452689413053958, 4.316510491518577, 4.572975431529405, 5.298024605638520, 5.815721586477977, 6.580975420045259, 6.815353993429146, 7.492166044330213, 8.009942880801348, 8.492755620147226, 8.959412507287587, 9.583913557100016, 10.10068521046238, 10.45556947511850, 11.12429844038056, 11.51330289910840, 11.99816803273836, 12.67351147871960, 12.84178556049644, 13.45165215699206, 13.93517665015018
