| L(s) = 1 | + i·2-s − 4-s + (2 − i)5-s − i·7-s − i·8-s + (1 + 2i)10-s + 4i·13-s + 14-s + 16-s − 2i·17-s + 8·19-s + (−2 + i)20-s − 8i·23-s + (3 − 4i)25-s − 4·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.377i·7-s − 0.353i·8-s + (0.316 + 0.632i)10-s + 1.10i·13-s + 0.267·14-s + 0.250·16-s − 0.485i·17-s + 1.83·19-s + (−0.447 + 0.223i)20-s − 1.66i·23-s + (0.600 − 0.800i)25-s − 0.784·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64819 + 0.389086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64819 + 0.389086i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−10.28407581988392245336394221870, −9.748709981893755748827259663325, −8.871692668389388938023742599383, −8.092437036545450606450883741632, −6.85270210861731788899936460411, −6.37258544847786355920627883817, −5.09090846982921197241254871485, −4.52203207450589931304947774670, −2.89000593295966485594978736565, −1.19424583573088466237925119486,
1.35996701240880706455183564253, 2.72103708481317336112328359056, 3.52409716305951997408299658200, 5.22957916698797422259489754979, 5.67725156428613387770483250389, 6.97825568682782207222386409766, 8.029479390995284509235287394639, 9.036282714935392713844527018644, 9.941061038410829275019050461368, 10.31529745828327732583015865858
