| L(s) = 1 | + 3-s + (−1.73 + 1.41i)5-s + 9-s − 3.46·13-s + (−1.73 + 1.41i)15-s − 4.89i·17-s − 4.89i·19-s − 2.82i·23-s + (0.999 − 4.89i)25-s + 27-s − 2.82i·29-s + 6.92·31-s + 3.46·37-s − 3.46·39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (−0.774 + 0.632i)5-s + 0.333·9-s − 0.960·13-s + (−0.447 + 0.365i)15-s − 1.18i·17-s − 1.12i·19-s − 0.589i·23-s + (0.199 − 0.979i)25-s + 0.192·27-s − 0.525i·29-s + 1.24·31-s + 0.569·37-s − 0.554·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.535727000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535727000\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (1.73 - 1.41i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 9.79iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 9.79iT - 97T^{2} \) |
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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
−9.081411460768140461935745184132, −8.262359173068766865677164319234, −7.35916754585065638224527834738, −7.11162295224853154724432222560, −6.02147177071246781574084235364, −4.71392721003970153822049750684, −4.23239430940952964300168075348, −2.82903340256090084427092583307, −2.58910120981751912731098264335, −0.58012374767346486552981182021,
1.18644393444132064850103467035, 2.41842727414296273319561657365, 3.61484183489885115418729493274, 4.22429063894540174337600183467, 5.16648501916682998411122317954, 6.11751051589626308615555579718, 7.23087856018087705070798270876, 7.87511815774549233715037790213, 8.414143609546948410211324497153, 9.219470234711872986937547732714
