L(s) = 1 | − 0.375·2-s + 3-s − 1.85·4-s + 1.27·5-s − 0.375·6-s + 2.77·7-s + 1.44·8-s + 9-s − 0.479·10-s + 0.396·11-s − 1.85·12-s − 5.77·13-s − 1.04·14-s + 1.27·15-s + 3.17·16-s − 17-s − 0.375·18-s − 4.30·19-s − 2.37·20-s + 2.77·21-s − 0.149·22-s + 4.90·23-s + 1.44·24-s − 3.36·25-s + 2.16·26-s + 27-s − 5.16·28-s + ⋯ |
L(s) = 1 | − 0.265·2-s + 0.577·3-s − 0.929·4-s + 0.571·5-s − 0.153·6-s + 1.04·7-s + 0.512·8-s + 0.333·9-s − 0.151·10-s + 0.119·11-s − 0.536·12-s − 1.60·13-s − 0.278·14-s + 0.329·15-s + 0.793·16-s − 0.242·17-s − 0.0884·18-s − 0.988·19-s − 0.531·20-s + 0.605·21-s − 0.0317·22-s + 1.02·23-s + 0.295·24-s − 0.673·25-s + 0.424·26-s + 0.192·27-s − 0.975·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 0.375T + 2T^{2} \) |
| 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 0.396T + 11T^{2} \) |
| 13 | \( 1 + 5.77T + 13T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 + 6.53T + 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 2.21T + 83T^{2} \) |
| 89 | \( 1 + 9.04T + 89T^{2} \) |
| 97 | \( 1 - 19.1T + 97T^{2} \) |
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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
−7.57524543635584836706797535716, −7.10715391114327012583016796081, −6.02202165468385284991625951177, −5.15285437792640864149683926451, −4.69403531308547613702529553124, −4.10675480739695827345550774582, −2.98255498654455115553163813772, −2.10953514221798185759483537616, −1.39264554737674375749785648090, 0,
1.39264554737674375749785648090, 2.10953514221798185759483537616, 2.98255498654455115553163813772, 4.10675480739695827345550774582, 4.69403531308547613702529553124, 5.15285437792640864149683926451, 6.02202165468385284991625951177, 7.10715391114327012583016796081, 7.57524543635584836706797535716