| L(s) = 1 | + 1.23i·5-s + 1.74i·7-s + 3.23i·11-s + (−2.23 + 2.82i)13-s − 5.65·17-s − 7.40i·19-s − 9.15·23-s + 3.47·25-s + 3.49·29-s − 1.74i·31-s − 2.16·35-s + 5.65i·37-s + 7.70i·41-s − 8.94·43-s + 7.23i·47-s + ⋯ |
| L(s) = 1 | + 0.552i·5-s + 0.660i·7-s + 0.975i·11-s + (−0.620 + 0.784i)13-s − 1.37·17-s − 1.69i·19-s − 1.90·23-s + 0.694·25-s + 0.649·29-s − 0.313i·31-s − 0.365·35-s + 0.929i·37-s + 1.20i·41-s − 1.36·43-s + 1.05i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.285459 + 0.821369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.285459 + 0.821369i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (2.23 - 2.82i)T \) |
| good | 5 | \( 1 - 1.23iT - 5T^{2} \) |
| 7 | \( 1 - 1.74iT - 7T^{2} \) |
| 11 | \( 1 - 3.23iT - 11T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 7.40iT - 19T^{2} \) |
| 23 | \( 1 + 9.15T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 1.74iT - 31T^{2} \) |
| 37 | \( 1 - 5.65iT - 37T^{2} \) |
| 41 | \( 1 - 7.70iT - 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 7.23iT - 47T^{2} \) |
| 53 | \( 1 + 9.15T + 53T^{2} \) |
| 59 | \( 1 - 1.70iT - 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 + 3.90iT - 67T^{2} \) |
| 71 | \( 1 - 5.70iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 0.944T + 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 - 10.7iT - 89T^{2} \) |
| 97 | \( 1 - 18.3iT - 97T^{2} \) |
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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
−10.28859018088739462429432185979, −9.549195850206311335906217317781, −8.840689414685057698858141651325, −7.84419179566455265384967481406, −6.75169679386214979705890033442, −6.44977424848073585847285419541, −4.92383429234176721042701639305, −4.36778179717875174315829992354, −2.77720336803026016565979184311, −2.04905929159133717275239352985,
0.38162500275537819878509770622, 1.96054267864831815759109510926, 3.44233217933916998914263906599, 4.30389880532893721297063991758, 5.39193739999270582897075815000, 6.20711754467214762829475662406, 7.25376719026253193977362541133, 8.241519930229718137387999691932, 8.654204654565732705764237373615, 9.941666030480093407969841536874
