| L(s) = 1 | − 2·7-s + 13-s − 3·17-s − 8·19-s + 3·23-s − 5·25-s + 9·29-s − 8·31-s + 8·37-s + 6·41-s + 43-s + 12·47-s − 3·49-s + 3·53-s + 6·59-s − 7·61-s − 14·67-s − 12·71-s + 2·73-s + 79-s − 12·89-s − 2·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.277·13-s − 0.727·17-s − 1.83·19-s + 0.625·23-s − 25-s + 1.67·29-s − 1.43·31-s + 1.31·37-s + 0.937·41-s + 0.152·43-s + 1.75·47-s − 3/7·49-s + 0.412·53-s + 0.781·59-s − 0.896·61-s − 1.71·67-s − 1.42·71-s + 0.234·73-s + 0.112·79-s − 1.27·89-s − 0.209·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.22900208286236, −12.73912369277374, −12.34156893891947, −11.85165763297166, −11.19658085425400, −10.82651805297040, −10.47234499924729, −9.942904351387206, −9.341977147274917, −8.981919419950658, −8.557761084253903, −8.073603638821914, −7.309287578725597, −7.096742754776333, −6.342616521826371, −6.043866535349670, −5.716631429063139, −4.758498669722487, −4.339635172773183, −3.999330465781136, −3.274912582875251, −2.633613447479532, −2.264426772503973, −1.506898505087583, −0.6761538932744648, 0,
0.6761538932744648, 1.506898505087583, 2.264426772503973, 2.633613447479532, 3.274912582875251, 3.999330465781136, 4.339635172773183, 4.758498669722487, 5.716631429063139, 6.043866535349670, 6.342616521826371, 7.096742754776333, 7.309287578725597, 8.073603638821914, 8.557761084253903, 8.981919419950658, 9.341977147274917, 9.942904351387206, 10.47234499924729, 10.82651805297040, 11.19658085425400, 11.85165763297166, 12.34156893891947, 12.73912369277374, 13.22900208286236
