L(s) = 1 | + (0.809 − 0.587i)2-s + (0.587 − 1.80i)3-s + (0.309 − 0.951i)4-s + (0.587 + 0.809i)5-s + (−0.587 − 1.80i)6-s − 7-s + (−0.309 − 0.951i)8-s + (−2.11 − 1.53i)9-s + (0.951 + 0.309i)10-s + (−1.53 − 1.11i)12-s + (1.53 + 1.11i)13-s + (−0.809 + 0.587i)14-s + (1.80 − 0.587i)15-s + (−0.809 − 0.587i)16-s − 2.61·18-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)2-s + (0.587 − 1.80i)3-s + (0.309 − 0.951i)4-s + (0.587 + 0.809i)5-s + (−0.587 − 1.80i)6-s − 7-s + (−0.309 − 0.951i)8-s + (−2.11 − 1.53i)9-s + (0.951 + 0.309i)10-s + (−1.53 − 1.11i)12-s + (1.53 + 1.11i)13-s + (−0.809 + 0.587i)14-s + (1.80 − 0.587i)15-s + (−0.809 − 0.587i)16-s − 2.61·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.986708233\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.986708233\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−9.368360794422079297157468901571, −8.788105069585683854899635143544, −7.49368946153719790522327226212, −6.73335676785600093140450833537, −6.27251791977765633352836181669, −5.76626750029523254191930381684, −3.82758548451803870063428620526, −3.12310501507340961386727973010, −2.22733079856664695251102372414, −1.38374054779614435746599674834,
2.58009072561909172004609424629, 3.49943729842409964607718231947, 4.04464906417913555312352078416, 5.05265426323449021838160194734, 5.71211956715592853784800289183, 6.34797930372937140818152126426, 7.963416621633534348222356356783, 8.572617869065944339923566441674, 9.142975688834912539235947074836, 9.997903716332180566735336122638