| L(s) = 1 | + 3.91·3-s − 7.82·5-s − 17.9·7-s − 11.6·9-s − 43.9·11-s − 16.3·13-s − 30.6·15-s − 103.·17-s − 131.·19-s − 70.3·21-s − 19.3·23-s − 63.7·25-s − 151.·27-s + 29·29-s + 164.·31-s − 172.·33-s + 140.·35-s + 299.·37-s − 64.0·39-s + 309.·41-s − 6.49·43-s + 91.0·45-s + 197.·47-s − 20.4·49-s − 405.·51-s + 402.·53-s + 343.·55-s + ⋯ |
| L(s) = 1 | + 0.754·3-s − 0.699·5-s − 0.969·7-s − 0.430·9-s − 1.20·11-s − 0.348·13-s − 0.527·15-s − 1.47·17-s − 1.58·19-s − 0.731·21-s − 0.175·23-s − 0.510·25-s − 1.07·27-s + 0.185·29-s + 0.954·31-s − 0.908·33-s + 0.678·35-s + 1.33·37-s − 0.263·39-s + 1.17·41-s − 0.0230·43-s + 0.301·45-s + 0.613·47-s − 0.0596·49-s − 1.11·51-s + 1.04·53-s + 0.843·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5146393666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5146393666\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 3.91T + 27T^{2} \) |
| 5 | \( 1 + 7.82T + 125T^{2} \) |
| 7 | \( 1 + 17.9T + 343T^{2} \) |
| 11 | \( 1 + 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.3T + 1.21e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.49T + 7.95e4T^{2} \) |
| 47 | \( 1 - 197.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 402.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 512.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 796.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 976.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 327.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 335.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 430.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 665.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 721.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.738799540232807154128489306873, −8.187222594876971839658099045213, −7.48711439396655701711556884740, −6.53439685505112898938711991370, −5.80566247306501829135217047052, −4.52339959864708406470686477118, −3.89315887883968566366603753992, −2.68073083130751797643316584745, −2.39338605115262030429146474536, −0.29383738637756856336311619772,
0.29383738637756856336311619772, 2.39338605115262030429146474536, 2.68073083130751797643316584745, 3.89315887883968566366603753992, 4.52339959864708406470686477118, 5.80566247306501829135217047052, 6.53439685505112898938711991370, 7.48711439396655701711556884740, 8.187222594876971839658099045213, 8.738799540232807154128489306873
