| L(s) = 1 | − 3·7-s + 3·13-s + 8·17-s − 5·19-s + 8·29-s + 3·31-s + 6·37-s − 4·41-s − 7·43-s − 4·47-s + 2·49-s + 4·53-s + 4·59-s + 5·61-s − 13·67-s + 8·71-s + 10·73-s − 4·79-s − 12·89-s − 9·91-s − 3·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 24·119-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.832·13-s + 1.94·17-s − 1.14·19-s + 1.48·29-s + 0.538·31-s + 0.986·37-s − 0.624·41-s − 1.06·43-s − 0.583·47-s + 2/7·49-s + 0.549·53-s + 0.520·59-s + 0.640·61-s − 1.58·67-s + 0.949·71-s + 1.17·73-s − 0.450·79-s − 1.27·89-s − 0.943·91-s − 0.304·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 2.20·119-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
| \(L(1)\) |
\(\approx\) |
\(2.304498757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304498757\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−13.61853684131221, −13.08628911365015, −12.79564274801141, −12.20894236998400, −11.82551347106453, −11.30235807155051, −10.48134117848535, −10.30712359366012, −9.756364540916490, −9.407834310096349, −8.514761952254731, −8.347781212763030, −7.803814161903581, −6.990995104145402, −6.584869543676676, −6.155869338731987, −5.663846770500200, −5.021766464120047, −4.332733291749142, −3.752392060168722, −3.164059692143865, −2.846885907874752, −1.926765111439868, −1.132018745481463, −0.5264792106738387,
0.5264792106738387, 1.132018745481463, 1.926765111439868, 2.846885907874752, 3.164059692143865, 3.752392060168722, 4.332733291749142, 5.021766464120047, 5.663846770500200, 6.155869338731987, 6.584869543676676, 6.990995104145402, 7.803814161903581, 8.347781212763030, 8.514761952254731, 9.407834310096349, 9.756364540916490, 10.30712359366012, 10.48134117848535, 11.30235807155051, 11.82551347106453, 12.20894236998400, 12.79564274801141, 13.08628911365015, 13.61853684131221
