L(s) = 1 | + 2-s + (−1.63 − 0.558i)3-s + 4-s + 5-s + (−1.63 − 0.558i)6-s − 2.20i·7-s + 8-s + (2.37 + 1.83i)9-s + 10-s − 6.11·11-s + (−1.63 − 0.558i)12-s + 3.23i·13-s − 2.20i·14-s + (−1.63 − 0.558i)15-s + 16-s + 4.87i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.946 − 0.322i)3-s + 0.5·4-s + 0.447·5-s + (−0.669 − 0.228i)6-s − 0.832i·7-s + 0.353·8-s + (0.791 + 0.610i)9-s + 0.316·10-s − 1.84·11-s + (−0.473 − 0.161i)12-s + 0.897i·13-s − 0.588i·14-s + (−0.423 − 0.144i)15-s + 0.250·16-s + 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.395575335\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.395575335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.63 + 0.558i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (0.386 + 8.17i)T \) |
good | 7 | \( 1 + 2.20iT - 7T^{2} \) |
| 11 | \( 1 + 6.11T + 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 - 4.87iT - 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 - 5.78iT - 23T^{2} \) |
| 29 | \( 1 - 0.744iT - 29T^{2} \) |
| 31 | \( 1 - 6.99iT - 31T^{2} \) |
| 37 | \( 1 - 1.30T + 37T^{2} \) |
| 41 | \( 1 - 7.92T + 41T^{2} \) |
| 43 | \( 1 - 1.47iT - 43T^{2} \) |
| 47 | \( 1 + 0.615iT - 47T^{2} \) |
| 53 | \( 1 + 6.82T + 53T^{2} \) |
| 59 | \( 1 + 8.06iT - 59T^{2} \) |
| 61 | \( 1 - 3.78iT - 61T^{2} \) |
| 71 | \( 1 - 12.4iT - 71T^{2} \) |
| 73 | \( 1 - 5.03T + 73T^{2} \) |
| 79 | \( 1 - 6.67iT - 79T^{2} \) |
| 83 | \( 1 + 2.20iT - 83T^{2} \) |
| 89 | \( 1 - 15.6iT - 89T^{2} \) |
| 97 | \( 1 + 8.34iT - 97T^{2} \) |
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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
−9.600252739336396712858503295409, −8.224256412906709027563224052920, −7.51840536765732338071134321697, −6.82686204822840808465675367722, −6.02923092639903505591152483759, −5.32157116752326452319043497654, −4.63378355768485572626196140093, −3.69477224712548351692042174318, −2.36143217271430295995530908376, −1.35141054993190165734566402443,
0.43751650316260042810765168702, 2.34767759499959319921680973557, 2.93470713235905606075232047415, 4.40922314313280893335463567102, 5.10588084225172605235001751457, 5.67436368786037929308432371624, 6.21024647490120872696464771735, 7.32260829524243106807007089794, 8.028380894805085068286138366439, 9.157616651391924881936970382296