| L(s) = 1 | − 2.37·2-s − 3-s + 3.66·4-s + 0.734·5-s + 2.37·6-s + 2.72·7-s − 3.95·8-s + 9-s − 1.74·10-s − 6.23·11-s − 3.66·12-s + 2.96·13-s − 6.48·14-s − 0.734·15-s + 2.09·16-s + 17-s − 2.37·18-s + 0.0641·19-s + 2.69·20-s − 2.72·21-s + 14.8·22-s + 3.60·23-s + 3.95·24-s − 4.46·25-s − 7.05·26-s − 27-s + 9.98·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.577·3-s + 1.83·4-s + 0.328·5-s + 0.971·6-s + 1.03·7-s − 1.39·8-s + 0.333·9-s − 0.552·10-s − 1.88·11-s − 1.05·12-s + 0.822·13-s − 1.73·14-s − 0.189·15-s + 0.523·16-s + 0.242·17-s − 0.560·18-s + 0.0147·19-s + 0.601·20-s − 0.594·21-s + 3.16·22-s + 0.752·23-s + 0.808·24-s − 0.892·25-s − 1.38·26-s − 0.192·27-s + 1.88·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7699086115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7699086115\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.37T + 2T^{2} \) |
| 5 | \( 1 - 0.734T + 5T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 + 6.23T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 19 | \( 1 - 0.0641T + 19T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 - 2.69T + 37T^{2} \) |
| 41 | \( 1 - 5.63T + 41T^{2} \) |
| 43 | \( 1 + 0.575T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 + 6.60T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 - 7.14T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 8.92T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 4.20T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.908532587335042398964344829386, −7.58443055743753182256753029456, −6.50888943556680329984524137798, −6.04674770728479280241445746157, −5.04347361803664821672025535144, −4.63580241562496645032667920125, −3.09885590170570808319238616165, −2.29234311806808744225963672186, −1.44059434191109420343632500589, −0.61972900119227574639040709388,
0.61972900119227574639040709388, 1.44059434191109420343632500589, 2.29234311806808744225963672186, 3.09885590170570808319238616165, 4.63580241562496645032667920125, 5.04347361803664821672025535144, 6.04674770728479280241445746157, 6.50888943556680329984524137798, 7.58443055743753182256753029456, 7.908532587335042398964344829386
