L(s) = 1 | − 2·2-s + 4·4-s + 18·5-s − 8·8-s − 36·10-s + 72·11-s + 34·13-s + 16·16-s + 6·17-s − 92·19-s + 72·20-s − 144·22-s + 180·23-s + 199·25-s − 68·26-s + 114·29-s − 56·31-s − 32·32-s − 12·34-s − 34·37-s + 184·38-s − 144·40-s + 6·41-s + 164·43-s + 288·44-s − 360·46-s + 168·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.60·5-s − 0.353·8-s − 1.13·10-s + 1.97·11-s + 0.725·13-s + 1/4·16-s + 0.0856·17-s − 1.11·19-s + 0.804·20-s − 1.39·22-s + 1.63·23-s + 1.59·25-s − 0.512·26-s + 0.729·29-s − 0.324·31-s − 0.176·32-s − 0.0605·34-s − 0.151·37-s + 0.785·38-s − 0.569·40-s + 0.0228·41-s + 0.581·43-s + 0.986·44-s − 1.15·46-s + 0.521·47-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
\(L(2)\) |
\(\approx\) |
\(2.563854619\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.563854619\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 72 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 - 114 T + p^{3} T^{2} \) |
| 31 | \( 1 + 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 654 T + p^{3} T^{2} \) |
| 59 | \( 1 + 492 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 124 T + p^{3} T^{2} \) |
| 71 | \( 1 + 36 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1010 T + p^{3} T^{2} \) |
| 79 | \( 1 - 56 T + p^{3} T^{2} \) |
| 83 | \( 1 - 228 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
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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
−9.407118957599589777774002170685, −9.200056436410743497429356186856, −8.408185828921642917801582607053, −6.91066912777679626291211004295, −6.44194210392446888012988099487, −5.71025986585677697841032687952, −4.38284179730316682386577635708, −3.04056704126805008078311618872, −1.77051363424117812309554853089, −1.09970433370330253512085902862,
1.09970433370330253512085902862, 1.77051363424117812309554853089, 3.04056704126805008078311618872, 4.38284179730316682386577635708, 5.71025986585677697841032687952, 6.44194210392446888012988099487, 6.91066912777679626291211004295, 8.408185828921642917801582607053, 9.200056436410743497429356186856, 9.407118957599589777774002170685