| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 3·11-s + 3·13-s − 15-s − 5·19-s − 21-s + 4·23-s − 4·25-s + 27-s − 5·29-s + 2·31-s + 3·33-s + 35-s − 10·37-s + 3·39-s − 45-s + 3·47-s − 6·49-s − 4·53-s − 3·55-s − 5·57-s + 4·59-s − 8·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.832·13-s − 0.258·15-s − 1.14·19-s − 0.218·21-s + 0.834·23-s − 4/5·25-s + 0.192·27-s − 0.928·29-s + 0.359·31-s + 0.522·33-s + 0.169·35-s − 1.64·37-s + 0.480·39-s − 0.149·45-s + 0.437·47-s − 6/7·49-s − 0.549·53-s − 0.404·55-s − 0.662·57-s + 0.520·59-s − 1.02·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.963983640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963983640\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.50194301848167, −12.26411232727732, −11.52553427841549, −11.19911468624406, −10.79777661291407, −10.27716902912785, −9.686366869063905, −9.295288758078380, −8.905060329775917, −8.381420027516326, −8.132174846133865, −7.464945213297957, −6.972319769971851, −6.549591869831752, −6.183402922296541, −5.531322794018910, −4.963299745260965, −4.284458876126607, −3.909685563516793, −3.492850459688424, −3.071077159555481, −2.259522875343785, −1.751712101260908, −1.203688630304777, −0.3533345448021691,
0.3533345448021691, 1.203688630304777, 1.751712101260908, 2.259522875343785, 3.071077159555481, 3.492850459688424, 3.909685563516793, 4.284458876126607, 4.963299745260965, 5.531322794018910, 6.183402922296541, 6.549591869831752, 6.972319769971851, 7.464945213297957, 8.132174846133865, 8.381420027516326, 8.905060329775917, 9.295288758078380, 9.686366869063905, 10.27716902912785, 10.79777661291407, 11.19911468624406, 11.52553427841549, 12.26411232727732, 12.50194301848167
