| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s − 4·11-s − 13-s − 2·14-s + 16-s − 4·17-s − 2·19-s − 20-s − 4·22-s − 2·23-s + 25-s − 26-s − 2·28-s − 8·29-s + 4·31-s + 32-s − 4·34-s + 2·35-s + 6·37-s − 2·38-s − 40-s − 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s − 0.223·20-s − 0.852·22-s − 0.417·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 1.48·29-s + 0.718·31-s + 0.176·32-s − 0.685·34-s + 0.338·35-s + 0.986·37-s − 0.324·38-s − 0.158·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.482781510226594161874514821285, −8.402306645073389208875709046616, −7.61851966333210959386324957870, −6.77872728355853698929699351273, −5.96022304844885243033851457877, −4.99762985109706213804877258036, −4.13007375875743923718905749312, −3.12318249922503105552823930899, −2.18442355690526819133910690087, 0,
2.18442355690526819133910690087, 3.12318249922503105552823930899, 4.13007375875743923718905749312, 4.99762985109706213804877258036, 5.96022304844885243033851457877, 6.77872728355853698929699351273, 7.61851966333210959386324957870, 8.402306645073389208875709046616, 9.482781510226594161874514821285
