| L(s) = 1 | + 0.0147·3-s − 2.88·5-s − 1.61·7-s − 2.99·9-s − 3.67·11-s − 5.42·13-s − 0.0427·15-s − 0.739·17-s − 2.07·19-s − 0.0238·21-s − 2.75·23-s + 3.34·25-s − 0.0887·27-s + 0.0289·29-s + 3.10·31-s − 0.0544·33-s + 4.65·35-s − 1.40·37-s − 0.0802·39-s − 11.0·41-s − 4.99·43-s + 8.66·45-s − 7.39·47-s − 4.39·49-s − 0.0109·51-s + 7.42·53-s + 10.6·55-s + ⋯ |
| L(s) = 1 | + 0.00854·3-s − 1.29·5-s − 0.609·7-s − 0.999·9-s − 1.10·11-s − 1.50·13-s − 0.0110·15-s − 0.179·17-s − 0.475·19-s − 0.00520·21-s − 0.574·23-s + 0.668·25-s − 0.0170·27-s + 0.00537·29-s + 0.558·31-s − 0.00947·33-s + 0.787·35-s − 0.230·37-s − 0.0128·39-s − 1.72·41-s − 0.762·43-s + 1.29·45-s − 1.07·47-s − 0.628·49-s − 0.00153·51-s + 1.02·53-s + 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1106971600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1106971600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0147T + 3T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 13 | \( 1 + 5.42T + 13T^{2} \) |
| 17 | \( 1 + 0.739T + 17T^{2} \) |
| 19 | \( 1 + 2.07T + 19T^{2} \) |
| 23 | \( 1 + 2.75T + 23T^{2} \) |
| 29 | \( 1 - 0.0289T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 4.99T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 - 7.42T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 3.31T + 61T^{2} \) |
| 67 | \( 1 - 0.0846T + 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 + 7.88T + 73T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.258482942130399638791518115435, −7.85162250492992126715859529376, −7.06968516333574870919868043441, −6.35152950951521172159129095098, −5.27804450661870117049552614782, −4.77483403074290392995138593187, −3.72675590097632123251096688748, −3.00079788946848226611600919105, −2.24026126125597698334440119736, −0.17338979827958018822831528729,
0.17338979827958018822831528729, 2.24026126125597698334440119736, 3.00079788946848226611600919105, 3.72675590097632123251096688748, 4.77483403074290392995138593187, 5.27804450661870117049552614782, 6.35152950951521172159129095098, 7.06968516333574870919868043441, 7.85162250492992126715859529376, 8.258482942130399638791518115435
