| L(s) = 1 | + 1.53·5-s + (0.414 + 2.61i)7-s + 0.585i·11-s − 2.16i·13-s + 5.86·17-s + 5.22i·19-s − 2.24i·23-s − 2.65·25-s + 5.41i·29-s + 4.32i·31-s + (0.634 + 4i)35-s + 4·37-s + 8.92·41-s − 10.4·43-s + 7.39·47-s + ⋯ |
| L(s) = 1 | + 0.684·5-s + (0.156 + 0.987i)7-s + 0.176i·11-s − 0.600i·13-s + 1.42·17-s + 1.19i·19-s − 0.467i·23-s − 0.531·25-s + 1.00i·29-s + 0.777i·31-s + (0.107 + 0.676i)35-s + 0.657·37-s + 1.39·41-s − 1.59·43-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.849320869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.849320869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.414 - 2.61i)T \) |
| good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 11 | \( 1 - 0.585iT - 11T^{2} \) |
| 13 | \( 1 + 2.16iT - 13T^{2} \) |
| 17 | \( 1 - 5.86T + 17T^{2} \) |
| 19 | \( 1 - 5.22iT - 19T^{2} \) |
| 23 | \( 1 + 2.24iT - 23T^{2} \) |
| 29 | \( 1 - 5.41iT - 29T^{2} \) |
| 31 | \( 1 - 4.32iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 4.58iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 5.86T + 89T^{2} \) |
| 97 | \( 1 - 8.28iT - 97T^{2} \) |
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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
−9.995284318137745449485199200834, −9.358410007113588816493611612292, −8.359910925662010180838426424666, −7.74739716749173901900777506191, −6.50567586669958736008308275172, −5.64492539769875528808664121786, −5.17854964115624286983590436641, −3.67878395041024523805485258077, −2.61507622180717270589630644302, −1.45918243763939807438092904076,
0.950517270968322911644530638124, 2.30371925162194826674678194064, 3.61347517446041846520424856331, 4.55858252784311560724558724901, 5.61023679040729220766239367263, 6.43610956678431681940129061410, 7.42205675973586739405604242610, 8.034388470771734565791941853429, 9.338053058729107558744817523501, 9.732823644615020455147102759473
