| L(s) = 1 | + 2·7-s − 4·13-s + 3·17-s + 4·19-s − 3·23-s − 31-s − 2·37-s + 6·41-s + 8·43-s + 3·47-s − 3·49-s + 9·53-s − 12·59-s − 5·61-s − 2·67-s − 12·71-s + 8·73-s + 79-s − 6·89-s − 8·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.10·13-s + 0.727·17-s + 0.917·19-s − 0.625·23-s − 0.179·31-s − 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.437·47-s − 3/7·49-s + 1.23·53-s − 1.56·59-s − 0.640·61-s − 0.244·67-s − 1.42·71-s + 0.936·73-s + 0.112·79-s − 0.635·89-s − 0.838·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.550·119-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
| \(L(1)\) |
\(\approx\) |
\(2.426170138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.426170138\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.80406465257853, −13.24018017621355, −12.52266600358575, −12.05727932671494, −11.98658809502750, −11.19608032211017, −10.74348802251318, −10.30066214429551, −9.593167593426304, −9.410792317810762, −8.732486096983860, −8.067300101647064, −7.624133869625199, −7.382045620236314, −6.717115380684542, −5.897893720357066, −5.565709791795916, −5.005374015564439, −4.424667740979954, −3.956706721641442, −3.107288942397719, −2.653301099088389, −1.908542108951058, −1.292926726387494, −0.4970505668610241,
0.4970505668610241, 1.292926726387494, 1.908542108951058, 2.653301099088389, 3.107288942397719, 3.956706721641442, 4.424667740979954, 5.005374015564439, 5.565709791795916, 5.897893720357066, 6.717115380684542, 7.382045620236314, 7.624133869625199, 8.067300101647064, 8.732486096983860, 9.410792317810762, 9.593167593426304, 10.30066214429551, 10.74348802251318, 11.19608032211017, 11.98658809502750, 12.05727932671494, 12.52266600358575, 13.24018017621355, 13.80406465257853
