| L(s) = 1 | − 2·2-s + 4·4-s + 14·7-s − 8·8-s + 22·11-s − 30·13-s − 28·14-s + 16·16-s + 7·17-s − 81·19-s − 44·22-s + 151·23-s + 60·26-s + 56·28-s − 270·29-s − 113·31-s − 32·32-s − 14·34-s − 88·37-s + 162·38-s + 406·41-s − 442·43-s + 88·44-s − 302·46-s + 56·47-s − 147·49-s − 120·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.603·11-s − 0.640·13-s − 0.534·14-s + 1/4·16-s + 0.0998·17-s − 0.978·19-s − 0.426·22-s + 1.36·23-s + 0.452·26-s + 0.377·28-s − 1.72·29-s − 0.654·31-s − 0.176·32-s − 0.0706·34-s − 0.391·37-s + 0.691·38-s + 1.54·41-s − 1.56·43-s + 0.301·44-s − 0.967·46-s + 0.173·47-s − 3/7·49-s − 0.320·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 - 7 T + p^{3} T^{2} \) |
| 19 | \( 1 + 81 T + p^{3} T^{2} \) |
| 23 | \( 1 - 151 T + p^{3} T^{2} \) |
| 29 | \( 1 + 270 T + p^{3} T^{2} \) |
| 31 | \( 1 + 113 T + p^{3} T^{2} \) |
| 37 | \( 1 + 88 T + p^{3} T^{2} \) |
| 41 | \( 1 - 406 T + p^{3} T^{2} \) |
| 43 | \( 1 + 442 T + p^{3} T^{2} \) |
| 47 | \( 1 - 56 T + p^{3} T^{2} \) |
| 53 | \( 1 + 141 T + p^{3} T^{2} \) |
| 59 | \( 1 + 274 T + p^{3} T^{2} \) |
| 61 | \( 1 - 41 T + p^{3} T^{2} \) |
| 67 | \( 1 - 328 T + p^{3} T^{2} \) |
| 71 | \( 1 + 390 T + p^{3} T^{2} \) |
| 73 | \( 1 - 626 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1215 T + p^{3} T^{2} \) |
| 83 | \( 1 - 505 T + p^{3} T^{2} \) |
| 89 | \( 1 - 514 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1816 T + p^{3} T^{2} \) |
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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
−8.967766789374293997535730368412, −8.038662546562051904702822172928, −7.33683127314068681861151532648, −6.57546341011011425777806781833, −5.51642747044117009804475986098, −4.62063369769631504940951836279, −3.50648589909764473815600874668, −2.24212943014126405007408390725, −1.35503176020619252481124025633, 0,
1.35503176020619252481124025633, 2.24212943014126405007408390725, 3.50648589909764473815600874668, 4.62063369769631504940951836279, 5.51642747044117009804475986098, 6.57546341011011425777806781833, 7.33683127314068681861151532648, 8.038662546562051904702822172928, 8.967766789374293997535730368412
