| L(s) = 1 | + 2.60·2-s + 2.13·3-s + 4.80·4-s + 5.58·6-s + 0.988·7-s + 7.30·8-s + 1.57·9-s + 10.2·12-s + 3.07·13-s + 2.57·14-s + 9.44·16-s − 1.61·17-s + 4.11·18-s − 6.58·19-s + 2.11·21-s + 5.18·23-s + 15.6·24-s + 8.02·26-s − 3.04·27-s + 4.74·28-s − 7.31·29-s − 2.64·31-s + 10.0·32-s − 4.22·34-s + 7.57·36-s + 2.80·37-s − 17.1·38-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 1.23·3-s + 2.40·4-s + 2.27·6-s + 0.373·7-s + 2.58·8-s + 0.526·9-s + 2.96·12-s + 0.853·13-s + 0.689·14-s + 2.36·16-s − 0.392·17-s + 0.970·18-s − 1.51·19-s + 0.461·21-s + 1.08·23-s + 3.18·24-s + 1.57·26-s − 0.585·27-s + 0.897·28-s − 1.35·29-s − 0.474·31-s + 1.77·32-s − 0.724·34-s + 1.26·36-s + 0.460·37-s − 2.78·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.237991030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.237991030\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 3 | \( 1 - 2.13T + 3T^{2} \) |
| 7 | \( 1 - 0.988T + 7T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 5.18T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 + 9.84T + 67T^{2} \) |
| 71 | \( 1 + 0.243T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 7.69T + 79T^{2} \) |
| 83 | \( 1 + 5.95T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
| 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
−8.604384696911231321174570062545, −7.84306396910461681611360970540, −7.05760938997117504285794762624, −6.30230479757443327385242597784, −5.54027424195385879640721408092, −4.62748268394033134646411371553, −3.90793537673939563416339690063, −3.31981783614053753210947166591, −2.40886903362086258846199863503, −1.73614283978036890288984322805,
1.73614283978036890288984322805, 2.40886903362086258846199863503, 3.31981783614053753210947166591, 3.90793537673939563416339690063, 4.62748268394033134646411371553, 5.54027424195385879640721408092, 6.30230479757443327385242597784, 7.05760938997117504285794762624, 7.84306396910461681611360970540, 8.604384696911231321174570062545
