| L(s) = 1 | − 3·7-s − 3·11-s + 13-s − 3·17-s + 4·23-s − 5·29-s + 3·31-s − 12·37-s − 2·41-s − 4·43-s + 3·47-s + 2·49-s − 9·53-s + 15·59-s − 3·61-s + 7·67-s − 8·71-s − 16·73-s + 9·77-s − 83-s − 3·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.834·23-s − 0.928·29-s + 0.538·31-s − 1.97·37-s − 0.312·41-s − 0.609·43-s + 0.437·47-s + 2/7·49-s − 1.23·53-s + 1.95·59-s − 0.384·61-s + 0.855·67-s − 0.949·71-s − 1.87·73-s + 1.02·77-s − 0.109·83-s − 0.314·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5608411497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5608411497\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.65581598105188, −13.96877228255504, −13.43213934492724, −13.02828248721885, −12.77845737949295, −12.05186022886064, −11.48579288783815, −10.93290484856217, −10.36172731465351, −9.999082686821208, −9.383661161931275, −8.738877099229175, −8.478345190628132, −7.575317172708595, −7.113491188199512, −6.586905619255510, −6.058845987729818, −5.329303965385391, −4.923621137740518, −4.060976691773393, −3.427677301036444, −2.931318653587426, −2.234684611167610, −1.398392584638774, −0.2620265356973018,
0.2620265356973018, 1.398392584638774, 2.234684611167610, 2.931318653587426, 3.427677301036444, 4.060976691773393, 4.923621137740518, 5.329303965385391, 6.058845987729818, 6.586905619255510, 7.113491188199512, 7.575317172708595, 8.478345190628132, 8.738877099229175, 9.383661161931275, 9.999082686821208, 10.36172731465351, 10.93290484856217, 11.48579288783815, 12.05186022886064, 12.77845737949295, 13.02828248721885, 13.43213934492724, 13.96877228255504, 14.65581598105188
