L(s) = 1 | + 2.51·2-s + 2.78·3-s + 4.34·4-s + 1.47·5-s + 7.01·6-s + 5.91·8-s + 4.75·9-s + 3.72·10-s − 4.83·11-s + 12.1·12-s − 4.02·13-s + 4.11·15-s + 6.20·16-s − 2.82·17-s + 11.9·18-s + 4.32·19-s + 6.42·20-s − 12.1·22-s − 7.27·23-s + 16.4·24-s − 2.81·25-s − 10.1·26-s + 4.89·27-s + 7.18·29-s + 10.3·30-s + 7.03·31-s + 3.81·32-s + ⋯ |
L(s) = 1 | + 1.78·2-s + 1.60·3-s + 2.17·4-s + 0.660·5-s + 2.86·6-s + 2.09·8-s + 1.58·9-s + 1.17·10-s − 1.45·11-s + 3.49·12-s − 1.11·13-s + 1.06·15-s + 1.55·16-s − 0.684·17-s + 2.82·18-s + 0.992·19-s + 1.43·20-s − 2.59·22-s − 1.51·23-s + 3.36·24-s − 0.563·25-s − 1.98·26-s + 0.941·27-s + 1.33·29-s + 1.89·30-s + 1.26·31-s + 0.673·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.725501009\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.725501009\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 - 2.78T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 + 4.02T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 31 | \( 1 - 7.03T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 43 | \( 1 - 0.174T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 - 7.57T + 53T^{2} \) |
| 59 | \( 1 - 3.13T + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.42T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−9.141358257522063873739736183443, −8.091691904166899872612985831100, −7.55724152347879021553350743882, −6.72086596976175418565921457057, −5.71314902501319008186946867729, −4.96921268403003883318966216271, −4.20111719799836857760235489419, −3.19640378516257667528037142488, −2.48502651514869324762712140082, −2.08504522787093778250763345268,
2.08504522787093778250763345268, 2.48502651514869324762712140082, 3.19640378516257667528037142488, 4.20111719799836857760235489419, 4.96921268403003883318966216271, 5.71314902501319008186946867729, 6.72086596976175418565921457057, 7.55724152347879021553350743882, 8.091691904166899872612985831100, 9.141358257522063873739736183443