| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 7-s + 8-s + 9-s − 3·10-s + 12-s − 3·13-s + 14-s − 3·15-s + 16-s − 5·17-s + 18-s − 8·19-s − 3·20-s + 21-s + 23-s + 24-s + 4·25-s − 3·26-s + 27-s + 28-s − 9·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s − 0.832·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 1.83·19-s − 0.670·20-s + 0.218·21-s + 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s + 0.188·28-s − 1.67·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.14422882850495, −13.56944883228842, −12.96206232437329, −12.77231351787071, −12.19514714355542, −11.76569005523008, −11.08418468999740, −10.91602171008228, −10.45918335181873, −9.541956966422258, −9.124100289131536, −8.502897658597893, −8.189892945467797, −7.436655331433079, −7.317801880782040, −6.657396904458047, −6.126457724708095, −5.294367510204521, −4.714325144843919, −4.391775095088004, −3.772437699887910, −3.470843994837580, −2.616248188483285, −2.051673745429528, −1.574029664043703, 0, 0,
1.574029664043703, 2.051673745429528, 2.616248188483285, 3.470843994837580, 3.772437699887910, 4.391775095088004, 4.714325144843919, 5.294367510204521, 6.126457724708095, 6.657396904458047, 7.317801880782040, 7.436655331433079, 8.189892945467797, 8.502897658597893, 9.124100289131536, 9.541956966422258, 10.45918335181873, 10.91602171008228, 11.08418468999740, 11.76569005523008, 12.19514714355542, 12.77231351787071, 12.96206232437329, 13.56944883228842, 14.14422882850495
