| L(s) = 1 | − 1.21·2-s − 1.04·3-s − 0.526·4-s + 2.40·5-s + 1.26·6-s − 0.534·7-s + 3.06·8-s − 1.91·9-s − 2.91·10-s − 0.714·11-s + 0.548·12-s − 5.63·13-s + 0.649·14-s − 2.50·15-s − 2.66·16-s + 4.81·17-s + 2.32·18-s + 2.70·19-s − 1.26·20-s + 0.556·21-s + 0.867·22-s + 23-s − 3.19·24-s + 0.772·25-s + 6.84·26-s + 5.11·27-s + 0.281·28-s + ⋯ |
| L(s) = 1 | − 0.858·2-s − 0.600·3-s − 0.263·4-s + 1.07·5-s + 0.515·6-s − 0.202·7-s + 1.08·8-s − 0.638·9-s − 0.922·10-s − 0.215·11-s + 0.158·12-s − 1.56·13-s + 0.173·14-s − 0.645·15-s − 0.667·16-s + 1.16·17-s + 0.548·18-s + 0.619·19-s − 0.282·20-s + 0.121·21-s + 0.184·22-s + 0.208·23-s − 0.651·24-s + 0.154·25-s + 1.34·26-s + 0.984·27-s + 0.0532·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 3 | \( 1 + 1.04T + 3T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 7 | \( 1 + 0.534T + 7T^{2} \) |
| 11 | \( 1 + 0.714T + 11T^{2} \) |
| 13 | \( 1 + 5.63T + 13T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 3.87T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 6.89T + 67T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 - 2.69T + 73T^{2} \) |
| 79 | \( 1 + 4.57T + 79T^{2} \) |
| 83 | \( 1 + 4.36T + 83T^{2} \) |
| 89 | \( 1 - 5.25T + 89T^{2} \) |
| 97 | \( 1 - 6.77T + 97T^{2} \) |
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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
−9.837516531344835236509117115140, −9.572065473743443234081563310132, −8.456355955455210741928193796520, −7.56865450484371503510442744428, −6.57430319471583996692913982854, −5.34253569269768512121410677751, −5.03209883472532142222176837057, −3.14409016534077726041113611391, −1.68668748856590489168862312243, 0,
1.68668748856590489168862312243, 3.14409016534077726041113611391, 5.03209883472532142222176837057, 5.34253569269768512121410677751, 6.57430319471583996692913982854, 7.56865450484371503510442744428, 8.456355955455210741928193796520, 9.572065473743443234081563310132, 9.837516531344835236509117115140
