| L(s) = 1 | − 2.70·5-s + (0.946 + 2.47i)7-s + 0.464i·11-s − 3.03i·13-s − 0.900·17-s − 0.831i·19-s − 7.12i·23-s + 2.33·25-s + 4.22i·29-s − 1.30i·31-s + (−2.56 − 6.69i)35-s + 10.3·37-s − 5.51·41-s + 11.8·43-s − 8.63·47-s + ⋯ |
| L(s) = 1 | − 1.21·5-s + (0.357 + 0.933i)7-s + 0.140i·11-s − 0.842i·13-s − 0.218·17-s − 0.190i·19-s − 1.48i·23-s + 0.466·25-s + 0.784i·29-s − 0.234i·31-s + (−0.433 − 1.13i)35-s + 1.70·37-s − 0.860·41-s + 1.80·43-s − 1.26·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.306315316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306315316\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.946 - 2.47i)T \) |
| good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 - 0.464iT - 11T^{2} \) |
| 13 | \( 1 + 3.03iT - 13T^{2} \) |
| 17 | \( 1 + 0.900T + 17T^{2} \) |
| 19 | \( 1 + 0.831iT - 19T^{2} \) |
| 23 | \( 1 + 7.12iT - 23T^{2} \) |
| 29 | \( 1 - 4.22iT - 29T^{2} \) |
| 31 | \( 1 + 1.30iT - 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 5.51T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 8.63T + 47T^{2} \) |
| 53 | \( 1 - 8.45iT - 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 3.27iT - 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 + 7.76iT - 71T^{2} \) |
| 73 | \( 1 - 4.83iT - 73T^{2} \) |
| 79 | \( 1 + 5.04T + 79T^{2} \) |
| 83 | \( 1 - 5.31T + 83T^{2} \) |
| 89 | \( 1 - 5.22T + 89T^{2} \) |
| 97 | \( 1 - 12.3iT - 97T^{2} \) |
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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
−8.580489352791660204106457614108, −8.087398696719829983752463408333, −7.44656172398063480157381113921, −6.51828120539836238608424133531, −5.68328110863071385705390353175, −4.80082024327217826939720676554, −4.14120219036959638543420764268, −3.08550967857014303975301134968, −2.30924621856386105049230904822, −0.73562319421716886607594677260,
0.65258723774628215767393936853, 1.89523129240823615708830330915, 3.29811531417900701803907666453, 4.03365242514426305878003587693, 4.50941010035164023812313520191, 5.57266496665455990280747238975, 6.63144472087559174307557039516, 7.29657031969394125579346577636, 7.892003757130957269764449692361, 8.444029561310224180735854490813
