| L(s) = 1 | + 0.874·2-s − 1.23·4-s − 0.236·5-s − 2.82·8-s − 0.206·10-s + 0.540·11-s − 0.874·13-s + 5·17-s + 4.03·19-s + 0.291·20-s + 0.472·22-s + 5.99·23-s − 4.94·25-s − 0.763·26-s − 8.61·29-s + 6.53·31-s + 5.65·32-s + 4.37·34-s + 8.70·37-s + 3.52·38-s + 0.667·40-s + 8.70·41-s − 2.23·43-s − 0.667·44-s + 5.23·46-s + 7.47·47-s − 4.32·50-s + ⋯ |
| L(s) = 1 | + 0.618·2-s − 0.618·4-s − 0.105·5-s − 0.999·8-s − 0.0652·10-s + 0.162·11-s − 0.242·13-s + 1.21·17-s + 0.925·19-s + 0.0652·20-s + 0.100·22-s + 1.24·23-s − 0.988·25-s − 0.149·26-s − 1.59·29-s + 1.17·31-s + 0.999·32-s + 0.749·34-s + 1.43·37-s + 0.572·38-s + 0.105·40-s + 1.35·41-s − 0.340·43-s − 0.100·44-s + 0.772·46-s + 1.08·47-s − 0.611·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.822085893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.822085893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.874T + 2T^{2} \) |
| 5 | \( 1 + 0.236T + 5T^{2} \) |
| 11 | \( 1 - 0.540T + 11T^{2} \) |
| 13 | \( 1 + 0.874T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 - 5.99T + 23T^{2} \) |
| 29 | \( 1 + 8.61T + 29T^{2} \) |
| 31 | \( 1 - 6.53T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 0.540T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 + 6.73T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.52T + 79T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 5.11T + 97T^{2} \) |
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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
−9.552037567740195741866037627306, −8.986942699617311809595778735524, −7.899605025888033944685304292187, −7.28499736460529922985217052010, −5.95935520872894079452128862703, −5.46405026093332529727973618499, −4.47880437122444323478276699536, −3.64649305687076378854128707365, −2.73156681447199638284421066344, −0.936863306241226625918730544336,
0.936863306241226625918730544336, 2.73156681447199638284421066344, 3.64649305687076378854128707365, 4.47880437122444323478276699536, 5.46405026093332529727973618499, 5.95935520872894079452128862703, 7.28499736460529922985217052010, 7.899605025888033944685304292187, 8.986942699617311809595778735524, 9.552037567740195741866037627306
