| L(s) = 1 | + 0.317i·5-s − i·7-s − 4.87·11-s − 2.44·13-s + 0.317i·17-s + 0.449i·19-s + 2.82·23-s + 4.89·25-s − 7.70i·29-s + 4.44i·31-s + 0.317·35-s − 7·37-s − 5.97i·41-s + 10.3i·43-s + 3.92·47-s + ⋯ |
| L(s) = 1 | + 0.142i·5-s − 0.377i·7-s − 1.47·11-s − 0.679·13-s + 0.0770i·17-s + 0.103i·19-s + 0.589·23-s + 0.979·25-s − 1.43i·29-s + 0.799i·31-s + 0.0537·35-s − 1.15·37-s − 0.933i·41-s + 1.57i·43-s + 0.572·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283635153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283635153\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.317iT - 5T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 0.317iT - 17T^{2} \) |
| 19 | \( 1 - 0.449iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 7.70iT - 29T^{2} \) |
| 31 | \( 1 - 4.44iT - 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 5.97iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 3.92T + 47T^{2} \) |
| 53 | \( 1 - 0.635iT - 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 9.34T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 - 8.55iT - 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 14.1iT - 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.006902121584950831580676065887, −7.52622107494010861897110120697, −6.84038949514523545602641494763, −6.05543572106895949911864293048, −5.08877101678570108428823323206, −4.78606936909720923991932420479, −3.65646825934433955684764814686, −2.84277115674060600909801294506, −2.12685481278895341299278794509, −0.76277133495505136671894290280,
0.43651893917590036738013617165, 1.83603037776665765298946941370, 2.71783255657289928031714124386, 3.33123160343807190673638077167, 4.57466292932010599667687462798, 5.15132441786490004059246942192, 5.62710050275022929255315806252, 6.70337070463327997514409767195, 7.30739714331350209735951412164, 7.934724654042935122970829938647
