| L(s) = 1 | − 5-s − 7-s − 3·9-s + 4·11-s − 6·13-s + 6·17-s + 19-s + 25-s − 6·29-s − 8·31-s + 35-s − 6·37-s − 6·41-s + 3·45-s + 49-s + 10·53-s − 4·55-s + 12·59-s + 2·61-s + 3·63-s + 6·65-s − 4·67-s − 12·71-s + 6·73-s − 4·77-s + 12·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.229·19-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.169·35-s − 0.986·37-s − 0.937·41-s + 0.447·45-s + 1/7·49-s + 1.37·53-s − 0.539·55-s + 1.56·59-s + 0.256·61-s + 0.377·63-s + 0.744·65-s − 0.488·67-s − 1.42·71-s + 0.702·73-s − 0.455·77-s + 1.35·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106606805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106606805\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−16.53306145157929, −16.30110070741927, −15.12190183277206, −14.78835987426163, −14.45452755128696, −13.84182341112852, −13.03272442962525, −12.31116296727782, −11.90145421610385, −11.61105892713578, −10.71779048909976, −10.04471504147951, −9.454442861734532, −8.956540813645537, −8.263090538759296, −7.368601628524886, −7.184868437880819, −6.257643959522296, −5.407093711593101, −5.129900244395653, −3.873596423915060, −3.524063185210880, −2.660779735230786, −1.718949178390209, −0.4803609141533560,
0.4803609141533560, 1.718949178390209, 2.660779735230786, 3.524063185210880, 3.873596423915060, 5.129900244395653, 5.407093711593101, 6.257643959522296, 7.184868437880819, 7.368601628524886, 8.263090538759296, 8.956540813645537, 9.454442861734532, 10.04471504147951, 10.71779048909976, 11.61105892713578, 11.90145421610385, 12.31116296727782, 13.03272442962525, 13.84182341112852, 14.45452755128696, 14.78835987426163, 15.12190183277206, 16.30110070741927, 16.53306145157929
