L(s) = 1 | − 1.48·5-s + 3.15·11-s + 6.76·13-s − 2·17-s + 1.03·19-s + 4.24·23-s − 2.79·25-s − 29-s + 1.87·31-s − 0.969·37-s + 7.52·41-s − 1.09·43-s − 9.34·47-s − 7·49-s − 5.73·53-s − 4.68·55-s + 8.24·59-s − 10.4·61-s − 10.0·65-s + 4.49·67-s + 10.1·71-s + 11.5·73-s + 13.0·79-s + 16.2·83-s + 2.96·85-s − 13.5·89-s − 1.52·95-s + ⋯ |
L(s) = 1 | − 0.664·5-s + 0.951·11-s + 1.87·13-s − 0.485·17-s + 0.236·19-s + 0.886·23-s − 0.559·25-s − 0.185·29-s + 0.336·31-s − 0.159·37-s + 1.17·41-s − 0.166·43-s − 1.36·47-s − 49-s − 0.787·53-s − 0.631·55-s + 1.07·59-s − 1.34·61-s − 1.24·65-s + 0.549·67-s + 1.20·71-s + 1.34·73-s + 1.47·79-s + 1.78·83-s + 0.322·85-s − 1.44·89-s − 0.156·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.930405733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.930405733\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 + 0.969T + 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 + 1.09T + 43T^{2} \) |
| 47 | \( 1 + 9.34T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.49T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 4.96T + 97T^{2} \) |
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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
−8.260738854880334246190818755418, −7.916222155992404520876827896358, −6.71478920997882914267176387218, −6.44960504187809977407494200763, −5.47799223374428180968531762040, −4.48935858807372632311785772787, −3.77076401327842547761460117824, −3.21437331220507161843547041853, −1.80163102510392151767884283348, −0.830705945922961809383246498234,
0.830705945922961809383246498234, 1.80163102510392151767884283348, 3.21437331220507161843547041853, 3.77076401327842547761460117824, 4.48935858807372632311785772787, 5.47799223374428180968531762040, 6.44960504187809977407494200763, 6.71478920997882914267176387218, 7.916222155992404520876827896358, 8.260738854880334246190818755418