| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s − 3·11-s + 3·13-s − 15-s − 17-s − 19-s − 4·21-s − 3·23-s − 4·25-s + 27-s − 10·29-s − 6·31-s − 3·33-s + 4·35-s − 4·37-s + 3·39-s + 5·41-s + 43-s − 45-s + 2·47-s + 9·49-s − 51-s − 14·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.904·11-s + 0.832·13-s − 0.258·15-s − 0.242·17-s − 0.229·19-s − 0.872·21-s − 0.625·23-s − 4/5·25-s + 0.192·27-s − 1.85·29-s − 1.07·31-s − 0.522·33-s + 0.676·35-s − 0.657·37-s + 0.480·39-s + 0.780·41-s + 0.152·43-s − 0.149·45-s + 0.291·47-s + 9/7·49-s − 0.140·51-s − 1.92·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−9.675525066681270992922543647705, −9.084920304961742227506252285555, −8.072177526566527517368179968812, −7.35280308552422744887681720550, −6.36108351595641399127151168680, −5.49242782557140989358879409039, −3.94051825844322199835949389881, −3.41153606626170658888682623087, −2.16737322822576147605235138245, 0,
2.16737322822576147605235138245, 3.41153606626170658888682623087, 3.94051825844322199835949389881, 5.49242782557140989358879409039, 6.36108351595641399127151168680, 7.35280308552422744887681720550, 8.072177526566527517368179968812, 9.084920304961742227506252285555, 9.675525066681270992922543647705
