L(s) = 1 | − 2.77·2-s + 5.71·4-s + 5-s − 2.71·7-s − 10.3·8-s − 2.77·10-s + 2.71·11-s + 13-s + 7.55·14-s + 17.2·16-s + 2.83·17-s − 3.55·19-s + 5.71·20-s − 7.55·22-s + 4.83·23-s + 25-s − 2.77·26-s − 15.5·28-s − 6·29-s + 7.55·31-s − 27.3·32-s − 7.88·34-s − 2.71·35-s − 4.27·37-s + 9.88·38-s − 10.3·40-s − 2.83·41-s + ⋯ |
L(s) = 1 | − 1.96·2-s + 2.85·4-s + 0.447·5-s − 1.02·7-s − 3.65·8-s − 0.878·10-s + 0.820·11-s + 0.277·13-s + 2.01·14-s + 4.31·16-s + 0.688·17-s − 0.816·19-s + 1.27·20-s − 1.61·22-s + 1.00·23-s + 0.200·25-s − 0.544·26-s − 2.93·28-s − 1.11·29-s + 1.35·31-s − 4.83·32-s − 1.35·34-s − 0.459·35-s − 0.703·37-s + 1.60·38-s − 1.63·40-s − 0.443·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6176510005\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6176510005\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 1.16T + 53T^{2} \) |
| 59 | \( 1 - 2.11T + 59T^{2} \) |
| 61 | \( 1 - 6.60T + 61T^{2} \) |
| 67 | \( 1 - 1.88T + 67T^{2} \) |
| 71 | \( 1 - 6.71T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 2.11T + 83T^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−10.43761722461859984654595425075, −9.618321240138132926479710099222, −9.139975139726218193931962025380, −8.331119901153000427704086331770, −7.20325079308604950179887002220, −6.54581875346392669929961587589, −5.78230755826575569186748267385, −3.51445313017621777717836226457, −2.33808916350099996578520064895, −0.933298169058262645999197157346,
0.933298169058262645999197157346, 2.33808916350099996578520064895, 3.51445313017621777717836226457, 5.78230755826575569186748267385, 6.54581875346392669929961587589, 7.20325079308604950179887002220, 8.331119901153000427704086331770, 9.139975139726218193931962025380, 9.618321240138132926479710099222, 10.43761722461859984654595425075