| L(s) = 1 | + 2·2-s + 2·3-s + 4·4-s + 4·6-s − 8·7-s + 8·8-s − 23·9-s + 44·11-s + 8·12-s − 16·14-s + 16·16-s + 74·17-s − 46·18-s + 19·19-s − 16·21-s + 88·22-s − 84·23-s + 16·24-s − 100·27-s − 32·28-s + 266·29-s + 136·31-s + 32·32-s + 88·33-s + 148·34-s − 92·36-s − 424·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.384·3-s + 1/2·4-s + 0.272·6-s − 0.431·7-s + 0.353·8-s − 0.851·9-s + 1.20·11-s + 0.192·12-s − 0.305·14-s + 1/4·16-s + 1.05·17-s − 0.602·18-s + 0.229·19-s − 0.166·21-s + 0.852·22-s − 0.761·23-s + 0.136·24-s − 0.712·27-s − 0.215·28-s + 1.70·29-s + 0.787·31-s + 0.176·32-s + 0.464·33-s + 0.746·34-s − 0.425·36-s − 1.88·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.845375495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.845375495\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 266 T + p^{3} T^{2} \) |
| 31 | \( 1 - 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 424 T + p^{3} T^{2} \) |
| 41 | \( 1 - 470 T + p^{3} T^{2} \) |
| 43 | \( 1 - 236 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 36 T + p^{3} T^{2} \) |
| 59 | \( 1 - 736 T + p^{3} T^{2} \) |
| 61 | \( 1 - 650 T + p^{3} T^{2} \) |
| 67 | \( 1 - 830 T + p^{3} T^{2} \) |
| 71 | \( 1 + 216 T + p^{3} T^{2} \) |
| 73 | \( 1 + 254 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1220 T + p^{3} T^{2} \) |
| 83 | \( 1 - 688 T + p^{3} T^{2} \) |
| 89 | \( 1 - 102 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1280 T + p^{3} T^{2} \) |
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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
−9.686546748191192949047116150339, −8.770323777938274341720523001103, −8.003556907006720126630005231341, −6.92631040859535809444346283162, −6.15516864450153892965121762221, −5.36585979044383558714186630595, −4.12743951225125155148413079329, −3.35806608768270544146569356912, −2.42379109088933040786155653206, −0.964209094043970544780360757486,
0.964209094043970544780360757486, 2.42379109088933040786155653206, 3.35806608768270544146569356912, 4.12743951225125155148413079329, 5.36585979044383558714186630595, 6.15516864450153892965121762221, 6.92631040859535809444346283162, 8.003556907006720126630005231341, 8.770323777938274341720523001103, 9.686546748191192949047116150339
