| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 13-s + 14-s + 16-s + 3·17-s − 2·19-s + 3·23-s − 5·25-s − 26-s − 28-s − 6·29-s − 4·31-s − 32-s − 3·34-s + 8·37-s + 2·38-s + 9·41-s + 4·43-s − 3·46-s + 49-s + 5·50-s + 52-s − 9·53-s + 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.625·23-s − 25-s − 0.196·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s + 1.31·37-s + 0.324·38-s + 1.40·41-s + 0.609·43-s − 0.442·46-s + 1/7·49-s + 0.707·50-s + 0.138·52-s − 1.23·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.79216720250414, −14.58285756956018, −13.91061838611795, −13.12681519460897, −12.87620468887485, −12.32870555983539, −11.64158668722310, −11.06521795400887, −10.88421920135183, −10.04716773752739, −9.511832507555760, −9.341358575781301, −8.543390159292054, −8.027987959674452, −7.473108420460588, −7.059913386770700, −6.262324967774952, −5.813105175658847, −5.338279828567368, −4.299342422551254, −3.850140319061346, −3.071546354872811, −2.448562010250656, −1.673923375366475, −0.9044856488413381, 0,
0.9044856488413381, 1.673923375366475, 2.448562010250656, 3.071546354872811, 3.850140319061346, 4.299342422551254, 5.338279828567368, 5.813105175658847, 6.262324967774952, 7.059913386770700, 7.473108420460588, 8.027987959674452, 8.543390159292054, 9.341358575781301, 9.511832507555760, 10.04716773752739, 10.88421920135183, 11.06521795400887, 11.64158668722310, 12.32870555983539, 12.87620468887485, 13.12681519460897, 13.91061838611795, 14.58285756956018, 14.79216720250414
