| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 7-s − 8-s + 9-s + 2·12-s − 4·13-s + 14-s + 16-s − 18-s + 2·19-s − 2·21-s − 2·24-s − 5·25-s + 4·26-s − 4·27-s − 28-s + 6·29-s + 4·31-s − 32-s + 36-s − 2·37-s − 2·38-s − 8·39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.577·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.436·21-s − 0.408·24-s − 25-s + 0.784·26-s − 0.769·27-s − 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1/6·36-s − 0.328·37-s − 0.324·38-s − 1.28·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.090068662745416752784621377032, −7.61488245805577994602083893658, −6.87015167727116618005628901277, −6.04516833074826431165424798565, −5.05791232432713970996781243183, −4.04379426959865020166490254054, −3.04116088637864497579983367990, −2.57969057241224021562840012921, −1.54182063036964932495656311078, 0,
1.54182063036964932495656311078, 2.57969057241224021562840012921, 3.04116088637864497579983367990, 4.04379426959865020166490254054, 5.05791232432713970996781243183, 6.04516833074826431165424798565, 6.87015167727116618005628901277, 7.61488245805577994602083893658, 8.090068662745416752784621377032
