| L(s) = 1 | + 4.29i·5-s + 7-s − 1.60i·11-s + 6.02i·13-s + 3.16·17-s − 3.07i·19-s − 5.95·23-s − 13.4·25-s − 3.53i·29-s + 2.08·31-s + 4.29i·35-s + 4.48i·37-s − 6.69·41-s + 12.2i·43-s − 3.82·47-s + ⋯ |
| L(s) = 1 | + 1.92i·5-s + 0.377·7-s − 0.484i·11-s + 1.67i·13-s + 0.766·17-s − 0.706i·19-s − 1.24·23-s − 2.69·25-s − 0.657i·29-s + 0.375·31-s + 0.726i·35-s + 0.736i·37-s − 1.04·41-s + 1.87i·43-s − 0.558·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9913185861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9913185861\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 - 4.29iT - 5T^{2} \) |
| 11 | \( 1 + 1.60iT - 11T^{2} \) |
| 13 | \( 1 - 6.02iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 3.07iT - 19T^{2} \) |
| 23 | \( 1 + 5.95T + 23T^{2} \) |
| 29 | \( 1 + 3.53iT - 29T^{2} \) |
| 31 | \( 1 - 2.08T + 31T^{2} \) |
| 37 | \( 1 - 4.48iT - 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 - 12.2iT - 43T^{2} \) |
| 47 | \( 1 + 3.82T + 47T^{2} \) |
| 53 | \( 1 + 8.63iT - 53T^{2} \) |
| 59 | \( 1 - 1.55iT - 59T^{2} \) |
| 61 | \( 1 - 3.64iT - 61T^{2} \) |
| 67 | \( 1 + 0.772iT - 67T^{2} \) |
| 71 | \( 1 + 7.36T + 71T^{2} \) |
| 73 | \( 1 - 1.32T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 7.08iT - 83T^{2} \) |
| 89 | \( 1 - 9.73T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.250679198900145183358552568425, −7.70545538826624015368226890925, −6.92649101314507527033616158105, −6.43028150671446867998671845214, −5.92379401276816156642368214446, −4.75551851728999968172725202583, −3.95534704786615221074404899903, −3.20832623657852278096746555361, −2.45137725997481175081819916836, −1.61724384461978845933420917491,
0.24775036261246403502097418460, 1.23905429539520531939281755259, 1.98900676912277765436168811362, 3.33500364936860033817315017719, 4.13989537874542364210223711791, 4.86801913535857542491901628326, 5.60389065680727395697357089842, 5.78449308973840259738936924050, 7.23016730282252131302248580393, 7.923772393409134749980423631914
