| L(s) = 1 | − 0.319·2-s + 2.34·3-s − 1.89·4-s + 1.26·5-s − 0.749·6-s + 1.17·7-s + 1.24·8-s + 2.48·9-s − 0.404·10-s + 1.82·11-s − 4.44·12-s − 2.29·13-s − 0.375·14-s + 2.96·15-s + 3.39·16-s + 1.44·17-s − 0.796·18-s + 6.82·19-s − 2.40·20-s + 2.75·21-s − 0.582·22-s + 23-s + 2.92·24-s − 3.39·25-s + 0.734·26-s − 1.19·27-s − 2.22·28-s + ⋯ |
| L(s) = 1 | − 0.226·2-s + 1.35·3-s − 0.948·4-s + 0.565·5-s − 0.306·6-s + 0.443·7-s + 0.440·8-s + 0.829·9-s − 0.127·10-s + 0.549·11-s − 1.28·12-s − 0.636·13-s − 0.100·14-s + 0.765·15-s + 0.849·16-s + 0.350·17-s − 0.187·18-s + 1.56·19-s − 0.536·20-s + 0.600·21-s − 0.124·22-s + 0.208·23-s + 0.596·24-s − 0.679·25-s + 0.144·26-s − 0.230·27-s − 0.421·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.011281015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011281015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.319T + 2T^{2} \) |
| 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 - 1.26T + 5T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 31 | \( 1 + 0.0171T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 8.65T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.22T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 1.95T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 8.10T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.50T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 5.50T + 97T^{2} \) |
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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
−9.990775790575130312174330668702, −9.446048080175507321668884362355, −9.029381333199901121933137967286, −7.87396297013429515977149165898, −7.57224929464010364228354676376, −5.92172162976422656387034485197, −4.87687727106816890935774041037, −3.84146295896125392803525916774, −2.76870276003720750198423075013, −1.39822304900822278672771261132,
1.39822304900822278672771261132, 2.76870276003720750198423075013, 3.84146295896125392803525916774, 4.87687727106816890935774041037, 5.92172162976422656387034485197, 7.57224929464010364228354676376, 7.87396297013429515977149165898, 9.029381333199901121933137967286, 9.446048080175507321668884362355, 9.990775790575130312174330668702
