| L(s) = 1 | + 5-s + 5·13-s − 5·17-s − 8·19-s + 4·23-s − 4·25-s + 3·29-s − 4·31-s + 3·37-s − 6·41-s − 4·43-s + 12·47-s − 7·49-s + 10·53-s − 8·59-s + 5·61-s + 5·65-s + 8·67-s − 16·71-s + 5·73-s − 4·79-s + 4·83-s − 5·85-s − 3·89-s − 8·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.38·13-s − 1.21·17-s − 1.83·19-s + 0.834·23-s − 4/5·25-s + 0.557·29-s − 0.718·31-s + 0.493·37-s − 0.937·41-s − 0.609·43-s + 1.75·47-s − 49-s + 1.37·53-s − 1.04·59-s + 0.640·61-s + 0.620·65-s + 0.977·67-s − 1.89·71-s + 0.585·73-s − 0.450·79-s + 0.439·83-s − 0.542·85-s − 0.317·89-s − 0.820·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990028644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990028644\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−13.90267563411973, −13.33764094490728, −13.22243300277618, −12.68836376472535, −12.02276516858359, −11.38200659913948, −11.02468143731798, −10.52759123283480, −10.17210976845132, −9.331275527372819, −8.931455670549542, −8.495266082730103, −8.129902643387646, −7.185269207110996, −6.795792471887600, −6.158582504431025, −5.933542599535868, −5.133075001806657, −4.464927688636913, −4.010654016654985, −3.414117431360098, −2.554737831223360, −2.026037895147185, −1.411609347709741, −0.4536316466710574,
0.4536316466710574, 1.411609347709741, 2.026037895147185, 2.554737831223360, 3.414117431360098, 4.010654016654985, 4.464927688636913, 5.133075001806657, 5.933542599535868, 6.158582504431025, 6.795792471887600, 7.185269207110996, 8.129902643387646, 8.495266082730103, 8.931455670549542, 9.331275527372819, 10.17210976845132, 10.52759123283480, 11.02468143731798, 11.38200659913948, 12.02276516858359, 12.68836376472535, 13.22243300277618, 13.33764094490728, 13.90267563411973
