L(s) = 1 | + (−0.587 − 0.809i)2-s + (−1.53 + 0.5i)3-s + (−0.309 + 0.951i)4-s + (1.30 + 0.951i)6-s + (0.951 − 0.309i)8-s + (1.30 − 0.951i)9-s + (0.309 + 0.951i)11-s − 1.61i·12-s + (−0.809 − 0.587i)16-s + (0.363 − 0.5i)17-s + (−1.53 − 0.5i)18-s + (−0.190 − 0.587i)19-s + (0.587 − 0.809i)22-s + (−1.30 + 0.951i)24-s + (−0.587 + 0.809i)27-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)2-s + (−1.53 + 0.5i)3-s + (−0.309 + 0.951i)4-s + (1.30 + 0.951i)6-s + (0.951 − 0.309i)8-s + (1.30 − 0.951i)9-s + (0.309 + 0.951i)11-s − 1.61i·12-s + (−0.809 − 0.587i)16-s + (0.363 − 0.5i)17-s + (−1.53 − 0.5i)18-s + (−0.190 − 0.587i)19-s + (0.587 − 0.809i)22-s + (−1.30 + 0.951i)24-s + (−0.587 + 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4690536121\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4690536121\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 1.61iT - T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - 0.618iT - T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - 1.61T + T^{2} \) |
| 97 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−9.519810374297941394276476832208, −8.830167463023455945662728366035, −7.62336274331124725272717512262, −6.98890393147669881754539250985, −6.11353019910224574862275620431, −5.01727474315674980783159404131, −4.53430961386149352302460351059, −3.58637988852881652985250836074, −2.25567563147923980347102741757, −0.913800761061847352398645822944,
0.70072901199509370151704436781, 1.79115313146577206944077807594, 3.70993273892511616409156033661, 4.89661783720719955001358140550, 5.54866431074480439498678062153, 6.23511525549115840545501808334, 6.64912463210591435461032501168, 7.59656099896304499974913655590, 8.273869127383854358035933911531, 9.116684102609768818994660615998