| L(s) = 1 | − 7-s − 4·11-s − 13-s + 4·29-s + 6·31-s − 12·37-s + 10·41-s + 8·43-s − 8·47-s + 49-s − 10·53-s − 4·59-s + 8·61-s + 2·67-s − 2·71-s + 6·73-s + 4·77-s − 8·79-s − 2·83-s + 18·89-s + 91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 0.277·13-s + 0.742·29-s + 1.07·31-s − 1.97·37-s + 1.56·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s − 0.520·59-s + 1.02·61-s + 0.244·67-s − 0.237·71-s + 0.702·73-s + 0.455·77-s − 0.900·79-s − 0.219·83-s + 1.90·89-s + 0.104·91-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−12.70717785514619, −12.45047423544222, −12.09055093990595, −11.35816508356968, −11.04104059123403, −10.47960889220943, −10.13329991275632, −9.759182236151472, −9.160837026328261, −8.730835399003184, −8.152691091831968, −7.801395246057736, −7.344737286814694, −6.757757753538433, −6.320771348884498, −5.832179710827152, −5.216369545827461, −4.866097682345661, −4.354912322136462, −3.671617914176788, −3.118624567838035, −2.639697212465209, −2.195523207910177, −1.406257853331664, −0.6660723299092157, 0,
0.6660723299092157, 1.406257853331664, 2.195523207910177, 2.639697212465209, 3.118624567838035, 3.671617914176788, 4.354912322136462, 4.866097682345661, 5.216369545827461, 5.832179710827152, 6.320771348884498, 6.757757753538433, 7.344737286814694, 7.801395246057736, 8.152691091831968, 8.730835399003184, 9.160837026328261, 9.759182236151472, 10.13329991275632, 10.47960889220943, 11.04104059123403, 11.35816508356968, 12.09055093990595, 12.45047423544222, 12.70717785514619
