L(s) = 1 | + i·3-s + (0.587 + 0.809i)5-s + (0.309 − 0.951i)7-s + (0.5 + 0.363i)11-s + (0.951 − 0.309i)13-s + (−0.809 + 0.587i)15-s + (−0.363 + 0.5i)17-s + (−0.951 − 0.309i)19-s + (0.951 + 0.309i)21-s + (−0.5 − 1.53i)23-s + i·27-s + (−1.30 + 0.951i)29-s + (0.951 − 1.30i)31-s + (−0.363 + 0.5i)33-s + (0.951 − 0.309i)35-s + ⋯ |
L(s) = 1 | + i·3-s + (0.587 + 0.809i)5-s + (0.309 − 0.951i)7-s + (0.5 + 0.363i)11-s + (0.951 − 0.309i)13-s + (−0.809 + 0.587i)15-s + (−0.363 + 0.5i)17-s + (−0.951 − 0.309i)19-s + (0.951 + 0.309i)21-s + (−0.5 − 1.53i)23-s + i·27-s + (−1.30 + 0.951i)29-s + (0.951 − 1.30i)31-s + (−0.363 + 0.5i)33-s + (0.951 − 0.309i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.252925742\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.252925742\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.951 - 0.309i)T \) |
good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 - 0.618iT - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + iT - T^{2} \) |
| 89 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−10.23228748423377008775007388546, −9.554415381699862557615579956319, −8.630223042284972389518924295001, −7.68590796738333211924994261685, −6.56387539676480124568784446204, −6.16659891511663796719533445890, −4.61656813550390256942635239426, −4.19860332853286735152006471386, −3.14445561347690398015634708095, −1.73059664554291718060909411395,
1.45851542342228874076480281989, 2.04776698668677107267462544796, 3.64648862564269952157790599424, 4.86823919753556064750794874024, 5.86519209612092490466493687100, 6.32994796539499533219158055566, 7.43693146472409404172443170018, 8.307642249710616038937211568633, 9.000522980282069315819318561953, 9.530743411663613486758947533379