| L(s) = 1 | − 4·11-s + 13-s + 2·17-s + 4·19-s + 6·23-s − 5·25-s + 6·29-s − 4·31-s − 8·37-s − 10·41-s − 8·43-s + 12·47-s + 2·53-s − 6·59-s − 10·61-s + 10·67-s + 8·71-s + 10·73-s + 8·79-s + 14·83-s + 18·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.277·13-s + 0.485·17-s + 0.917·19-s + 1.25·23-s − 25-s + 1.11·29-s − 0.718·31-s − 1.31·37-s − 1.56·41-s − 1.21·43-s + 1.75·47-s + 0.274·53-s − 0.781·59-s − 1.28·61-s + 1.22·67-s + 0.949·71-s + 1.17·73-s + 0.900·79-s + 1.53·83-s + 1.90·89-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.463446924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.463446924\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 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
−12.46376337802299, −12.08480130903092, −11.66581802754915, −11.11494459859311, −10.54836381208320, −10.41242638794704, −9.825582176667442, −9.360121585631976, −8.822363491451893, −8.424768872673577, −7.853244293123068, −7.523183064829514, −7.053967014790169, −6.486317945351022, −6.008995852792991, −5.272955805323195, −5.116858986946733, −4.740415898315824, −3.750051885055951, −3.399247170028253, −3.043365867415671, −2.222247957973738, −1.841837774408730, −0.9831873263222850, −0.4618218940142886,
0.4618218940142886, 0.9831873263222850, 1.841837774408730, 2.222247957973738, 3.043365867415671, 3.399247170028253, 3.750051885055951, 4.740415898315824, 5.116858986946733, 5.272955805323195, 6.008995852792991, 6.486317945351022, 7.053967014790169, 7.523183064829514, 7.853244293123068, 8.424768872673577, 8.822363491451893, 9.360121585631976, 9.825582176667442, 10.41242638794704, 10.54836381208320, 11.11494459859311, 11.66581802754915, 12.08480130903092, 12.46376337802299
