| L(s) = 1 | + 2·2-s + 4·4-s + 15·5-s + 8·8-s + 30·10-s + 9·11-s − 88·13-s + 16·16-s + 84·17-s + 104·19-s + 60·20-s + 18·22-s + 84·23-s + 100·25-s − 176·26-s − 51·29-s + 185·31-s + 32·32-s + 168·34-s + 44·37-s + 208·38-s + 120·40-s + 168·41-s + 326·43-s + 36·44-s + 168·46-s + 138·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 0.246·11-s − 1.87·13-s + 1/4·16-s + 1.19·17-s + 1.25·19-s + 0.670·20-s + 0.174·22-s + 0.761·23-s + 4/5·25-s − 1.32·26-s − 0.326·29-s + 1.07·31-s + 0.176·32-s + 0.847·34-s + 0.195·37-s + 0.887·38-s + 0.474·40-s + 0.639·41-s + 1.15·43-s + 0.123·44-s + 0.538·46-s + 0.428·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.482692324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.482692324\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 88 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 - 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 51 T + p^{3} T^{2} \) |
| 31 | \( 1 - 185 T + p^{3} T^{2} \) |
| 37 | \( 1 - 44 T + p^{3} T^{2} \) |
| 41 | \( 1 - 168 T + p^{3} T^{2} \) |
| 43 | \( 1 - 326 T + p^{3} T^{2} \) |
| 47 | \( 1 - 138 T + p^{3} T^{2} \) |
| 53 | \( 1 + 639 T + p^{3} T^{2} \) |
| 59 | \( 1 + 159 T + p^{3} T^{2} \) |
| 61 | \( 1 - 722 T + p^{3} T^{2} \) |
| 67 | \( 1 + 166 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 583 T + p^{3} T^{2} \) |
| 83 | \( 1 - 597 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 + 169 T + p^{3} T^{2} \) |
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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
−9.751482728702668631079545372529, −9.283385485332879576376611909445, −7.76066054603367320268103871182, −7.12063735095639332749051704323, −6.03530391305140994944387231524, −5.36575796157305422868752940391, −4.63698734568898660282238007202, −3.13766200021492677731824703999, −2.35338802470284293224795117524, −1.12036654108747367301933162029,
1.12036654108747367301933162029, 2.35338802470284293224795117524, 3.13766200021492677731824703999, 4.63698734568898660282238007202, 5.36575796157305422868752940391, 6.03530391305140994944387231524, 7.12063735095639332749051704323, 7.76066054603367320268103871182, 9.283385485332879576376611909445, 9.751482728702668631079545372529
