| L(s) = 1 | + 2.20·3-s − 0.114·5-s − 1.65·7-s + 1.86·9-s − 2.49·11-s + 4.47·13-s − 0.252·15-s − 5.86·17-s − 2.36·19-s − 3.65·21-s + 4.54·23-s − 4.98·25-s − 2.50·27-s + 4.58·29-s − 2.72·31-s − 5.49·33-s + 0.189·35-s + 5.48·37-s + 9.86·39-s − 2.63·41-s − 0.436·43-s − 0.213·45-s − 3.11·47-s − 4.25·49-s − 12.9·51-s − 7.34·53-s + 0.284·55-s + ⋯ |
| L(s) = 1 | + 1.27·3-s − 0.0511·5-s − 0.626·7-s + 0.621·9-s − 0.751·11-s + 1.24·13-s − 0.0651·15-s − 1.42·17-s − 0.543·19-s − 0.797·21-s + 0.946·23-s − 0.997·25-s − 0.482·27-s + 0.851·29-s − 0.489·31-s − 0.956·33-s + 0.0320·35-s + 0.902·37-s + 1.57·39-s − 0.411·41-s − 0.0665·43-s − 0.0317·45-s − 0.454·47-s − 0.607·49-s − 1.81·51-s − 1.00·53-s + 0.0384·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 2.20T + 3T^{2} \) |
| 5 | \( 1 + 0.114T + 5T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + 2.36T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 + 2.72T + 31T^{2} \) |
| 37 | \( 1 - 5.48T + 37T^{2} \) |
| 41 | \( 1 + 2.63T + 41T^{2} \) |
| 43 | \( 1 + 0.436T + 43T^{2} \) |
| 47 | \( 1 + 3.11T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 5.17T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 8.49T + 73T^{2} \) |
| 79 | \( 1 + 9.67T + 79T^{2} \) |
| 83 | \( 1 - 1.74T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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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
−8.008690783270611049623272089165, −7.05112099615025065761035180927, −6.42933821897445019514442329973, −5.68658222434026858949817452966, −4.59143955250947103406651391066, −3.89382155057312088374726953494, −3.10580060286407980629960045198, −2.53247447731105639171045459336, −1.58197556397522444184236805790, 0,
1.58197556397522444184236805790, 2.53247447731105639171045459336, 3.10580060286407980629960045198, 3.89382155057312088374726953494, 4.59143955250947103406651391066, 5.68658222434026858949817452966, 6.42933821897445019514442329973, 7.05112099615025065761035180927, 8.008690783270611049623272089165
