| L(s) = 1 | + 1.53·3-s + 3.41·5-s + 0.260·7-s − 0.640·9-s + 6.09·11-s + 6.30·13-s + 5.23·15-s − 0.137·17-s − 1.05·19-s + 0.400·21-s + 9.29·23-s + 6.63·25-s − 5.59·27-s + 0.622·29-s − 2.51·31-s + 9.35·33-s + 0.889·35-s + 0.830·37-s + 9.68·39-s − 8.31·41-s − 10.8·43-s − 2.18·45-s + 8.75·47-s − 6.93·49-s − 0.211·51-s + 12.6·53-s + 20.7·55-s + ⋯ |
| L(s) = 1 | + 0.886·3-s + 1.52·5-s + 0.0985·7-s − 0.213·9-s + 1.83·11-s + 1.74·13-s + 1.35·15-s − 0.0334·17-s − 0.241·19-s + 0.0874·21-s + 1.93·23-s + 1.32·25-s − 1.07·27-s + 0.115·29-s − 0.452·31-s + 1.62·33-s + 0.150·35-s + 0.136·37-s + 1.55·39-s − 1.29·41-s − 1.64·43-s − 0.325·45-s + 1.27·47-s − 0.990·49-s − 0.0296·51-s + 1.73·53-s + 2.80·55-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
| \(L(1)\) |
\(\approx\) |
\(4.697716443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.697716443\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.53T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.260T + 7T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 0.137T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 23 | \( 1 - 9.29T + 23T^{2} \) |
| 29 | \( 1 - 0.622T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 - 0.830T + 37T^{2} \) |
| 41 | \( 1 + 8.31T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−8.588532254926473849113630138292, −7.23865538886962271893874521024, −6.57270476946317738962172560005, −6.04181178601941925247092195720, −5.38439437737139469105157580583, −4.30235624136084190335092422820, −3.45282212478164144618760661893, −2.86059098536725337740686038253, −1.64288234283946670297055861962, −1.34085139183952287360822839855,
1.34085139183952287360822839855, 1.64288234283946670297055861962, 2.86059098536725337740686038253, 3.45282212478164144618760661893, 4.30235624136084190335092422820, 5.38439437737139469105157580583, 6.04181178601941925247092195720, 6.57270476946317738962172560005, 7.23865538886962271893874521024, 8.588532254926473849113630138292
