| L(s) = 1 | + 2-s + 4-s + (−1.82 − 1.29i)5-s + 2.99i·7-s + 8-s + (−1.82 − 1.29i)10-s − 1.08i·11-s − 1.37i·13-s + 2.99i·14-s + 16-s − 0.212·17-s + 6.70i·19-s + (−1.82 − 1.29i)20-s − 1.08i·22-s + 1.06i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.816 − 0.576i)5-s + 1.13i·7-s + 0.353·8-s + (−0.577 − 0.407i)10-s − 0.328i·11-s − 0.381i·13-s + 0.800i·14-s + 0.250·16-s − 0.0514·17-s + 1.53i·19-s + (−0.408 − 0.288i)20-s − 0.232i·22-s + 0.221i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.886457012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886457012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.82 + 1.29i)T \) |
| 29 | \( 1 + (2.49 + 4.77i)T \) |
| good | 7 | \( 1 - 2.99iT - 7T^{2} \) |
| 11 | \( 1 + 1.08iT - 11T^{2} \) |
| 13 | \( 1 + 1.37iT - 13T^{2} \) |
| 17 | \( 1 + 0.212T + 17T^{2} \) |
| 19 | \( 1 - 6.70iT - 19T^{2} \) |
| 23 | \( 1 - 1.06iT - 23T^{2} \) |
| 31 | \( 1 - 4.22iT - 31T^{2} \) |
| 37 | \( 1 + 5.72T + 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 + 3.38iT - 61T^{2} \) |
| 67 | \( 1 - 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 5.04T + 71T^{2} \) |
| 73 | \( 1 - 3.51T + 73T^{2} \) |
| 79 | \( 1 + 8.85iT - 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.95iT - 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.922960019397147196241117719235, −8.137975014671039107452576249092, −7.71897975572611820544963868921, −6.56914027391558388070911746418, −5.71591504735283936041342601664, −5.26387356971250220835435262189, −4.23685782487571271888063011386, −3.50497876090509752449305229702, −2.57193732775563004888614130040, −1.33428953038037950829302248859,
0.49676529042737460921118936047, 2.09757993340309384272631865177, 3.17420632972180342506595239864, 3.96888675068550113068417108554, 4.52127680805673555364951356541, 5.44686168139164730351752337424, 6.73681035426170823697806235378, 6.99670625982291139651170361401, 7.62046901648004695390587923251, 8.583703063141241780276365396311
