| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s + 7·7-s + 8·8-s + 10·10-s + 9·11-s − 19·13-s + 14·14-s + 16·16-s − 108·17-s + 11·19-s + 20·20-s + 18·22-s − 126·23-s + 25·25-s − 38·26-s + 28·28-s − 66·29-s − 148·31-s + 32·32-s − 216·34-s + 35·35-s − 346·37-s + 22·38-s + 40·40-s − 147·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.246·11-s − 0.405·13-s + 0.267·14-s + 1/4·16-s − 1.54·17-s + 0.132·19-s + 0.223·20-s + 0.174·22-s − 1.14·23-s + 1/5·25-s − 0.286·26-s + 0.188·28-s − 0.422·29-s − 0.857·31-s + 0.176·32-s − 1.08·34-s + 0.169·35-s − 1.53·37-s + 0.0939·38-s + 0.158·40-s − 0.559·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + 108 T + p^{3} T^{2} \) |
| 19 | \( 1 - 11 T + p^{3} T^{2} \) |
| 23 | \( 1 + 126 T + p^{3} T^{2} \) |
| 29 | \( 1 + 66 T + p^{3} T^{2} \) |
| 31 | \( 1 + 148 T + p^{3} T^{2} \) |
| 37 | \( 1 + 346 T + p^{3} T^{2} \) |
| 41 | \( 1 + 147 T + p^{3} T^{2} \) |
| 43 | \( 1 + 139 T + p^{3} T^{2} \) |
| 47 | \( 1 - 201 T + p^{3} T^{2} \) |
| 53 | \( 1 - 249 T + p^{3} T^{2} \) |
| 59 | \( 1 + 582 T + p^{3} T^{2} \) |
| 61 | \( 1 - 344 T + p^{3} T^{2} \) |
| 67 | \( 1 - 305 T + p^{3} T^{2} \) |
| 71 | \( 1 + 912 T + p^{3} T^{2} \) |
| 73 | \( 1 + 151 T + p^{3} T^{2} \) |
| 79 | \( 1 + 832 T + p^{3} T^{2} \) |
| 83 | \( 1 - 873 T + p^{3} T^{2} \) |
| 89 | \( 1 - 609 T + p^{3} T^{2} \) |
| 97 | \( 1 - 686 T + p^{3} T^{2} \) |
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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
−8.530954263026669586615088765746, −7.47718556543517602239525735227, −6.79939604676998135352814329857, −5.98442470505878768973832902842, −5.18564090909137158916049578537, −4.40368749013604261886425526161, −3.54227932212212552180626992194, −2.32204767100284632260022738102, −1.67025437963934209739767046565, 0,
1.67025437963934209739767046565, 2.32204767100284632260022738102, 3.54227932212212552180626992194, 4.40368749013604261886425526161, 5.18564090909137158916049578537, 5.98442470505878768973832902842, 6.79939604676998135352814329857, 7.47718556543517602239525735227, 8.530954263026669586615088765746
