| L(s) = 1 | + 2.21i·5-s − 1.26·7-s + i·11-s + 0.807i·13-s − 5.69·17-s − 5.87i·19-s + 7.89·23-s + 0.109·25-s + 7.17i·29-s − 7.31·31-s − 2.79i·35-s + 9.62i·37-s − 2.83·41-s + 2.83i·43-s − 7.22·47-s + ⋯ |
| L(s) = 1 | + 0.989i·5-s − 0.476·7-s + 0.301i·11-s + 0.223i·13-s − 1.38·17-s − 1.34i·19-s + 1.64·23-s + 0.0218·25-s + 1.33i·29-s − 1.31·31-s − 0.471i·35-s + 1.58i·37-s − 0.441·41-s + 0.431i·43-s − 1.05·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2603310685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2603310685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.21iT - 5T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 13 | \( 1 - 0.807iT - 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 5.87iT - 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 - 7.17iT - 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 9.62iT - 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 - 2.83iT - 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 - 3.01iT - 53T^{2} \) |
| 59 | \( 1 + 3.32iT - 59T^{2} \) |
| 61 | \( 1 + 8.97iT - 61T^{2} \) |
| 67 | \( 1 + 12.3iT - 67T^{2} \) |
| 71 | \( 1 + 2.56T + 71T^{2} \) |
| 73 | \( 1 + 9.01T + 73T^{2} \) |
| 79 | \( 1 + 2.35T + 79T^{2} \) |
| 83 | \( 1 + 2.50iT - 83T^{2} \) |
| 89 | \( 1 - 0.832T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−9.143961275177943787386192787737, −8.472114671643754663347450463927, −7.28395777938482516473924173960, −6.77401019331018690570620009372, −6.48766922099665054439502122432, −5.11901441801242899717778566332, −4.58073033694830168338757598764, −3.25513802973278408979311622425, −2.87387764517540450531264468144, −1.66266295422843400004231852248,
0.079350604941791020611071695099, 1.33351497644065391150139023824, 2.47198725085171343914044751634, 3.59893150929995508464716255342, 4.34011487271718494887528740538, 5.24815753552064567414989703830, 5.87445915078433756416004102817, 6.77256496182676360010217734861, 7.53077728784552082138812034482, 8.479494723729627918444376171349
