| L(s) = 1 | − 2·2-s + 4·3-s + 4·4-s − 8·6-s + 20·7-s − 8·8-s − 11·9-s − 44·11-s + 16·12-s − 42·13-s − 40·14-s + 16·16-s + 86·17-s + 22·18-s + 19·19-s + 80·21-s + 88·22-s + 164·23-s − 32·24-s + 84·26-s − 152·27-s + 80·28-s − 162·29-s − 312·31-s − 32·32-s − 176·33-s − 172·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.769·3-s + 1/2·4-s − 0.544·6-s + 1.07·7-s − 0.353·8-s − 0.407·9-s − 1.20·11-s + 0.384·12-s − 0.896·13-s − 0.763·14-s + 1/4·16-s + 1.22·17-s + 0.288·18-s + 0.229·19-s + 0.831·21-s + 0.852·22-s + 1.48·23-s − 0.272·24-s + 0.633·26-s − 1.08·27-s + 0.539·28-s − 1.03·29-s − 1.80·31-s − 0.176·32-s − 0.928·33-s − 0.867·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 86 T + p^{3} T^{2} \) |
| 23 | \( 1 - 164 T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 + 312 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 34 T + p^{3} T^{2} \) |
| 43 | \( 1 - 432 T + p^{3} T^{2} \) |
| 47 | \( 1 + 580 T + p^{3} T^{2} \) |
| 53 | \( 1 + 506 T + p^{3} T^{2} \) |
| 59 | \( 1 - 364 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 924 T + p^{3} T^{2} \) |
| 71 | \( 1 - 320 T + p^{3} T^{2} \) |
| 73 | \( 1 - 542 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1208 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1120 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1022 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1166 T + p^{3} T^{2} \) |
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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
−9.180700017115572303234957846163, −8.344635207839581109585745535152, −7.67907554782502979740626199614, −7.25183001897325441176490271790, −5.55243312343187576252414647160, −5.07098759768887744272554133966, −3.44182307120488631370652692612, −2.56605570032941137453342564380, −1.56649504698929036716723161686, 0,
1.56649504698929036716723161686, 2.56605570032941137453342564380, 3.44182307120488631370652692612, 5.07098759768887744272554133966, 5.55243312343187576252414647160, 7.25183001897325441176490271790, 7.67907554782502979740626199614, 8.344635207839581109585745535152, 9.180700017115572303234957846163
