| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 14-s + 16-s + 6·17-s + 4·19-s − 20-s + 25-s − 28-s + 6·29-s + 4·31-s + 32-s + 6·34-s + 35-s − 2·37-s + 4·38-s − 40-s + 6·41-s + 8·43-s − 12·47-s + 49-s + 50-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.141·50-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.07488958261698, −13.37384064269369, −13.03334803731372, −12.31093653276764, −12.11218926344957, −11.71579021803968, −11.10629673206171, −10.45318801135022, −10.21859432616815, −9.399204354312224, −9.232532968911253, −8.185991873312269, −7.941347889774111, −7.467220236806236, −6.789758929236968, −6.385027378271348, −5.673053576571916, −5.381333624346202, −4.558370027884461, −4.266922757241584, −3.430637091925425, −3.016676384359119, −2.658788306179820, −1.468460648140388, −1.080706558588881, 0,
1.080706558588881, 1.468460648140388, 2.658788306179820, 3.016676384359119, 3.430637091925425, 4.266922757241584, 4.558370027884461, 5.381333624346202, 5.673053576571916, 6.385027378271348, 6.789758929236968, 7.467220236806236, 7.941347889774111, 8.185991873312269, 9.232532968911253, 9.399204354312224, 10.21859432616815, 10.45318801135022, 11.10629673206171, 11.71579021803968, 12.11218926344957, 12.31093653276764, 13.03334803731372, 13.37384064269369, 14.07488958261698
