| L(s) = 1 | + (−1.65 + 0.517i)3-s − 5-s − 0.387i·7-s + (2.46 − 1.71i)9-s − 4.30·11-s − 3.73i·13-s + (1.65 − 0.517i)15-s + 3.85i·17-s − 5.95·19-s + (0.200 + 0.640i)21-s + 4.48i·23-s + 25-s + (−3.18 + 4.10i)27-s + 7.61i·29-s − 3.91i·31-s + ⋯ |
| L(s) = 1 | + (−0.954 + 0.298i)3-s − 0.447·5-s − 0.146i·7-s + (0.821 − 0.570i)9-s − 1.29·11-s − 1.03i·13-s + (0.426 − 0.133i)15-s + 0.934i·17-s − 1.36·19-s + (0.0437 + 0.139i)21-s + 0.934i·23-s + 0.200·25-s + (−0.613 + 0.789i)27-s + 1.41i·29-s − 0.702i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6739908855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6739908855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.65 - 0.517i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (7.70 + 2.75i)T \) |
| good | 7 | \( 1 + 0.387iT - 7T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 + 3.73iT - 13T^{2} \) |
| 17 | \( 1 - 3.85iT - 17T^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 23 | \( 1 - 4.48iT - 23T^{2} \) |
| 29 | \( 1 - 7.61iT - 29T^{2} \) |
| 31 | \( 1 + 3.91iT - 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 + 0.142T + 41T^{2} \) |
| 43 | \( 1 + 8.29iT - 43T^{2} \) |
| 47 | \( 1 - 0.257iT - 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 + 3.12iT - 59T^{2} \) |
| 61 | \( 1 - 8.47iT - 61T^{2} \) |
| 71 | \( 1 - 0.503iT - 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 + 2.98iT - 79T^{2} \) |
| 83 | \( 1 + 5.05iT - 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 - 3.95iT - 97T^{2} \) |
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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
−8.358937282606883871382989588468, −7.64383776527119411961324688094, −7.02155245976887705034669965441, −6.02255558996791961208673030464, −5.50945756653244380371548352567, −4.76774715609433973623180658126, −3.93076415929867082689281217027, −3.13402648425783212039998018651, −1.83836091273391058807030398132, −0.44702546402367756270806382490,
0.50992958666568299968912029403, 1.97958004038147601536147576424, 2.76583286590054150021518697558, 4.20850016013740083399410093584, 4.66555588567422388401229124043, 5.45441185608270078024658689694, 6.36952942812223556819778727892, 6.83408895053439406383457487994, 7.71635176689667888315124964489, 8.243155224549265568587788403222
