| L(s) = 1 | + 2-s − 4-s − 2·7-s − 3·8-s + 2·11-s − 13-s − 2·14-s − 16-s − 17-s + 2·22-s + 8·23-s − 5·25-s − 26-s + 2·28-s + 6·29-s + 6·31-s + 5·32-s − 34-s + 4·37-s + 12·41-s − 4·43-s − 2·44-s + 8·46-s − 3·49-s − 5·50-s + 52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.755·7-s − 1.06·8-s + 0.603·11-s − 0.277·13-s − 0.534·14-s − 1/4·16-s − 0.242·17-s + 0.426·22-s + 1.66·23-s − 25-s − 0.196·26-s + 0.377·28-s + 1.11·29-s + 1.07·31-s + 0.883·32-s − 0.171·34-s + 0.657·37-s + 1.87·41-s − 0.609·43-s − 0.301·44-s + 1.17·46-s − 3/7·49-s − 0.707·50-s + 0.138·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1989 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1989 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.782040728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782040728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.237634125983734665777444796591, −8.537674584116014643936461500198, −7.51652736314682879781148688875, −6.50892217077812457053338872148, −6.03783813983787978598423027694, −4.96254609148734960386106795551, −4.32046537490554466136211195238, −3.39280348926993285239486452160, −2.60550172422162141545165279017, −0.805347437926139300568670470568,
0.805347437926139300568670470568, 2.60550172422162141545165279017, 3.39280348926993285239486452160, 4.32046537490554466136211195238, 4.96254609148734960386106795551, 6.03783813983787978598423027694, 6.50892217077812457053338872148, 7.51652736314682879781148688875, 8.537674584116014643936461500198, 9.237634125983734665777444796591
