L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s + 16-s + 5.19·17-s − 6.92·19-s − 3·20-s + 4·25-s − 5.19·29-s + 6.92·31-s + 32-s + 5.19·34-s + 1.73·37-s − 6.92·38-s − 3·40-s − 9·41-s + 4·43-s − 12·47-s − 7·49-s + 4·50-s − 5.19·53-s − 5.19·58-s − 12·59-s − 5·61-s + 6.92·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 0.250·16-s + 1.26·17-s − 1.58·19-s − 0.670·20-s + 0.800·25-s − 0.964·29-s + 1.24·31-s + 0.176·32-s + 0.891·34-s + 0.284·37-s − 1.12·38-s − 0.474·40-s − 1.40·41-s + 0.609·43-s − 1.75·47-s − 49-s + 0.565·50-s − 0.713·53-s − 0.682·58-s − 1.56·59-s − 0.640·61-s + 0.879·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 8.66T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.970240796448642570782047959671, −7.80217389174266032176109521014, −6.72656485044841777917526710778, −6.12402114764985458034363130797, −5.04208262510933275150718684135, −4.38298962523770002752781485059, −3.62470125974034823452427526505, −2.93657157978365226984459472846, −1.59073177445135688052121487724, 0,
1.59073177445135688052121487724, 2.93657157978365226984459472846, 3.62470125974034823452427526505, 4.38298962523770002752781485059, 5.04208262510933275150718684135, 6.12402114764985458034363130797, 6.72656485044841777917526710778, 7.80217389174266032176109521014, 7.970240796448642570782047959671