| L(s) = 1 | − 2·3-s − 5-s + 7-s + 9-s + 4·11-s − 13-s + 2·15-s − 6·17-s + 19-s − 2·21-s + 23-s − 4·25-s + 4·27-s + 3·29-s − 7·31-s − 8·33-s − 35-s − 10·37-s + 2·39-s − 10·41-s − 7·43-s − 45-s − 9·47-s + 49-s + 12·51-s + 3·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.516·15-s − 1.45·17-s + 0.229·19-s − 0.436·21-s + 0.208·23-s − 4/5·25-s + 0.769·27-s + 0.557·29-s − 1.25·31-s − 1.39·33-s − 0.169·35-s − 1.64·37-s + 0.320·39-s − 1.56·41-s − 1.06·43-s − 0.149·45-s − 1.31·47-s + 1/7·49-s + 1.68·51-s + 0.412·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−10.17435826410628239370871594677, −9.061227760984532485276071768651, −8.329316103613891831727779509539, −7.00333273487530359010730637345, −6.54503125115405758625293787428, −5.39651500780218077097659142296, −4.61533440054158576539031736984, −3.55373972341149593647255533280, −1.73388343110170206846100508745, 0,
1.73388343110170206846100508745, 3.55373972341149593647255533280, 4.61533440054158576539031736984, 5.39651500780218077097659142296, 6.54503125115405758625293787428, 7.00333273487530359010730637345, 8.329316103613891831727779509539, 9.061227760984532485276071768651, 10.17435826410628239370871594677
