| L(s) = 1 | + 2.51·2-s + 4.32·4-s − 3.32·5-s + 4.51·7-s + 5.83·8-s − 8.34·10-s − 11-s − 1.32·13-s + 11.3·14-s + 6.02·16-s + 0.806·17-s − 6.34·19-s − 14.3·20-s − 2.51·22-s − 5.34·23-s + 6.02·25-s − 3.32·26-s + 19.5·28-s + 4.12·29-s − 1.32·31-s + 3.48·32-s + 2.02·34-s − 14.9·35-s − 1.09·37-s − 15.9·38-s − 19.3·40-s − 0.900·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.16·4-s − 1.48·5-s + 1.70·7-s + 2.06·8-s − 2.64·10-s − 0.301·11-s − 0.366·13-s + 3.03·14-s + 1.50·16-s + 0.195·17-s − 1.45·19-s − 3.20·20-s − 0.536·22-s − 1.11·23-s + 1.20·25-s − 0.651·26-s + 3.68·28-s + 0.766·29-s − 0.237·31-s + 0.616·32-s + 0.347·34-s − 2.53·35-s − 0.179·37-s − 2.58·38-s − 3.06·40-s − 0.140·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.030812906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.030812906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 0.806T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 - 4.12T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 + 0.900T + 41T^{2} \) |
| 43 | \( 1 + 2.22T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 - 9.73T + 59T^{2} \) |
| 61 | \( 1 + 8.93T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 5.67T + 71T^{2} \) |
| 73 | \( 1 - 0.386T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 + 3.96T + 83T^{2} \) |
| 89 | \( 1 - 9.41T + 89T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
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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
−11.96893192669411330401766109750, −11.25444728677071343466851953505, −10.57867881791384911493403369535, −8.346295976252343779134907841160, −7.79986970176634820348190492396, −6.71530107099102843450651599461, −5.30250464362511919292156769096, −4.49591440658217801100702389070, −3.81504358257950597324638431089, −2.21952300272845480830642440738,
2.21952300272845480830642440738, 3.81504358257950597324638431089, 4.49591440658217801100702389070, 5.30250464362511919292156769096, 6.71530107099102843450651599461, 7.79986970176634820348190492396, 8.346295976252343779134907841160, 10.57867881791384911493403369535, 11.25444728677071343466851953505, 11.96893192669411330401766109750
