| L(s) = 1 | + 5-s + 7-s − 2·11-s − 13-s + 2·17-s + 2·19-s + 8·23-s + 25-s + 35-s + 10·37-s − 10·41-s − 8·43-s + 49-s + 2·53-s − 2·55-s + 4·59-s − 8·61-s − 65-s + 6·67-s − 10·73-s − 2·77-s − 8·79-s − 2·83-s + 2·85-s − 18·89-s − 91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s − 0.277·13-s + 0.485·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s + 0.169·35-s + 1.64·37-s − 1.56·41-s − 1.21·43-s + 1/7·49-s + 0.274·53-s − 0.269·55-s + 0.520·59-s − 1.02·61-s − 0.124·65-s + 0.733·67-s − 1.17·73-s − 0.227·77-s − 0.900·79-s − 0.219·83-s + 0.216·85-s − 1.90·89-s − 0.104·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 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.s |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−14.61362183554351, −13.84652775035255, −13.48364042341606, −12.99323026530463, −12.57275205399634, −11.89843935350300, −11.36828906758054, −11.04633349220819, −10.23876336062036, −10.01042900005566, −9.428647776389698, −8.768722695715574, −8.384453486843900, −7.672892177950284, −7.279476171088258, −6.672497766886419, −6.070504512885270, −5.294782324549432, −5.121084692415246, −4.466982657146681, −3.646323235963986, −2.914291868185751, −2.570140666681057, −1.578529104154383, −1.082224922981173, 0,
1.082224922981173, 1.578529104154383, 2.570140666681057, 2.914291868185751, 3.646323235963986, 4.466982657146681, 5.121084692415246, 5.294782324549432, 6.070504512885270, 6.672497766886419, 7.279476171088258, 7.672892177950284, 8.384453486843900, 8.768722695715574, 9.428647776389698, 10.01042900005566, 10.23876336062036, 11.04633349220819, 11.36828906758054, 11.89843935350300, 12.57275205399634, 12.99323026530463, 13.48364042341606, 13.84652775035255, 14.61362183554351
