| L(s) = 1 | + 2-s + 2.60·3-s − 4-s + 3.76·5-s + 2.60·6-s + 0.167·7-s − 3·8-s + 3.76·9-s + 3.76·10-s + 11-s − 2.60·12-s + 0.167·14-s + 9.80·15-s − 16-s + 4.37·17-s + 3.76·18-s − 3.37·19-s − 3.76·20-s + 0.434·21-s + 22-s − 5.20·23-s − 7.80·24-s + 9.20·25-s + 2.00·27-s − 0.167·28-s + 1.93·29-s + 9.80·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.50·3-s − 0.5·4-s + 1.68·5-s + 1.06·6-s + 0.0631·7-s − 1.06·8-s + 1.25·9-s + 1.19·10-s + 0.301·11-s − 0.751·12-s + 0.0446·14-s + 2.53·15-s − 0.250·16-s + 1.05·17-s + 0.888·18-s − 0.773·19-s − 0.842·20-s + 0.0948·21-s + 0.213·22-s − 1.08·23-s − 1.59·24-s + 1.84·25-s + 0.384·27-s − 0.0315·28-s + 0.359·29-s + 1.79·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.732306576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.732306576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 7 | \( 1 - 0.167T + 7T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 - 6.86T + 31T^{2} \) |
| 37 | \( 1 - 8.63T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 + 7.20T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 + 4.80T + 61T^{2} \) |
| 67 | \( 1 + 4.13T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 3.56T + 97T^{2} \) |
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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
−9.407750181247093254240824399636, −8.415223831717115387485642193811, −8.062442496978053488398885439127, −6.58302359384078998077300083996, −6.02327506898658858389252976494, −5.09044550278763172614169077777, −4.20376448758559288954064020820, −3.22029146313498483787054389142, −2.50825995326670552262391988283, −1.50818878855248461912936282843,
1.50818878855248461912936282843, 2.50825995326670552262391988283, 3.22029146313498483787054389142, 4.20376448758559288954064020820, 5.09044550278763172614169077777, 6.02327506898658858389252976494, 6.58302359384078998077300083996, 8.062442496978053488398885439127, 8.415223831717115387485642193811, 9.407750181247093254240824399636
