L(s) = 1 | + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.955 + 0.294i)5-s + (0.222 + 0.974i)7-s + (0.222 − 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.142 − 1.90i)11-s + (0.658 − 0.317i)13-s + (0.365 − 0.930i)14-s + (−0.733 + 0.680i)16-s + (0.5 − 0.866i)18-s + (0.733 + 1.26i)19-s + (−0.623 − 0.781i)20-s + (−1.19 + 1.49i)22-s + (−0.988 − 0.149i)23-s + ⋯ |
L(s) = 1 | + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.955 + 0.294i)5-s + (0.222 + 0.974i)7-s + (0.222 − 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.142 − 1.90i)11-s + (0.658 − 0.317i)13-s + (0.365 − 0.930i)14-s + (−0.733 + 0.680i)16-s + (0.5 − 0.866i)18-s + (0.733 + 1.26i)19-s + (−0.623 − 0.781i)20-s + (−1.19 + 1.49i)22-s + (−0.988 − 0.149i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6471298285\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6471298285\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.826 + 0.563i)T \) |
| 5 | \( 1 + (0.955 - 0.294i)T \) |
| 7 | \( 1 + (-0.222 - 0.974i)T \) |
good | 3 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 11 | \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \) |
| 13 | \( 1 + (-0.658 + 0.317i)T + (0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 19 | \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.988 + 0.149i)T + (0.955 + 0.294i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.0546 - 0.139i)T + (-0.733 - 0.680i)T^{2} \) |
| 41 | \( 1 + (0.367 - 1.61i)T + (-0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-1.63 - 1.11i)T + (0.365 + 0.930i)T^{2} \) |
| 53 | \( 1 + (-0.722 - 1.84i)T + (-0.733 + 0.680i)T^{2} \) |
| 59 | \( 1 + (0.425 + 0.131i)T + (0.826 + 0.563i)T^{2} \) |
| 61 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.0931 - 1.24i)T + (-0.988 + 0.149i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.321583784283212541470681261240, −8.486388382495407235137028198097, −8.099291123699908595149286092560, −7.61852048170833764805851843628, −6.26044991535817353817187405275, −5.64891737706685746094408256646, −4.25723602648262526882715503653, −3.34220108149151363399831500896, −2.67755261693770355783166687736, −1.27008515873697204209095186142,
0.71809615588098880111143484022, 1.95342709877207399068346507918, 3.69249097145408699558380890039, 4.38814551299679949626899067382, 5.24457537502710194932914980318, 6.55550352568981600090809153166, 7.23098057089002254485400539018, 7.43379670647011800897532295687, 8.569142856019795253995971859439, 9.156922613286986473463480370520