| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·11-s + 13-s − 14-s + 16-s − 3·17-s − 4·19-s + 3·22-s + 3·23-s − 26-s + 28-s + 9·29-s − 10·31-s − 32-s + 3·34-s − 2·37-s + 4·38-s − 6·41-s + 43-s − 3·44-s − 3·46-s + 6·47-s − 6·49-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.917·19-s + 0.639·22-s + 0.625·23-s − 0.196·26-s + 0.188·28-s + 1.67·29-s − 1.79·31-s − 0.176·32-s + 0.514·34-s − 0.328·37-s + 0.648·38-s − 0.937·41-s + 0.152·43-s − 0.452·44-s − 0.442·46-s + 0.875·47-s − 6/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−15.31786915638917, −14.76394757549736, −14.25600622233782, −13.51134909542831, −13.10585574510195, −12.54870201155107, −11.97660067735205, −11.33846781411943, −10.80069343968846, −10.51098310839464, −9.983307856025706, −9.111072050918515, −8.783004488098544, −8.284085469895136, −7.702887139650387, −7.093788839399170, −6.573678492697037, −5.956865867357597, −5.168105334407402, −4.748532454635001, −3.885651128099238, −3.164696366117245, −2.354533847891303, −1.890570279343471, −0.8937035532183153, 0,
0.8937035532183153, 1.890570279343471, 2.354533847891303, 3.164696366117245, 3.885651128099238, 4.748532454635001, 5.168105334407402, 5.956865867357597, 6.573678492697037, 7.093788839399170, 7.702887139650387, 8.284085469895136, 8.783004488098544, 9.111072050918515, 9.983307856025706, 10.51098310839464, 10.80069343968846, 11.33846781411943, 11.97660067735205, 12.54870201155107, 13.10585574510195, 13.51134909542831, 14.25600622233782, 14.76394757549736, 15.31786915638917
