| L(s) = 1 | − 4·7-s + 5·13-s − 7·19-s + 11·31-s − 37-s + 5·43-s + 9·49-s − 61-s + 11·67-s − 7·73-s + 17·79-s − 20·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.38·13-s − 1.60·19-s + 1.97·31-s − 0.164·37-s + 0.762·43-s + 9/7·49-s − 0.128·61-s + 1.34·67-s − 0.819·73-s + 1.91·79-s − 2.09·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.745159526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745159526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.42897267094817, −13.26440868524781, −12.78139625150765, −12.27469920486531, −11.82727603606905, −11.12083429098131, −10.68764693709336, −10.25850115450287, −9.800688722765683, −9.150230531956780, −8.826713767048201, −8.223840472590451, −7.835802265433169, −6.884819308764844, −6.558339524772132, −6.209064743555771, −5.777078216322809, −4.955026014778313, −4.233477856890959, −3.850300821245604, −3.250525441112428, −2.671239394213814, −2.057167728065775, −1.101735722139818, −0.4556609994279276,
0.4556609994279276, 1.101735722139818, 2.057167728065775, 2.671239394213814, 3.250525441112428, 3.850300821245604, 4.233477856890959, 4.955026014778313, 5.777078216322809, 6.209064743555771, 6.558339524772132, 6.884819308764844, 7.835802265433169, 8.223840472590451, 8.826713767048201, 9.150230531956780, 9.800688722765683, 10.25850115450287, 10.68764693709336, 11.12083429098131, 11.82727603606905, 12.27469920486531, 12.78139625150765, 13.26440868524781, 13.42897267094817
