| L(s) = 1 | − 3.41·2-s − 3.08·3-s + 3.62·4-s − 5·5-s + 10.5·6-s − 22.1·7-s + 14.9·8-s − 17.4·9-s + 17.0·10-s − 32.6·11-s − 11.1·12-s − 13·13-s + 75.3·14-s + 15.4·15-s − 79.8·16-s − 26.8·17-s + 59.6·18-s − 73.2·19-s − 18.1·20-s + 68.1·21-s + 111.·22-s − 81.4·23-s − 45.9·24-s + 25·25-s + 44.3·26-s + 137.·27-s − 80.2·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 0.593·3-s + 0.453·4-s − 0.447·5-s + 0.715·6-s − 1.19·7-s + 0.658·8-s − 0.647·9-s + 0.539·10-s − 0.895·11-s − 0.269·12-s − 0.277·13-s + 1.43·14-s + 0.265·15-s − 1.24·16-s − 0.382·17-s + 0.780·18-s − 0.884·19-s − 0.202·20-s + 0.708·21-s + 1.08·22-s − 0.738·23-s − 0.390·24-s + 0.200·25-s + 0.334·26-s + 0.978·27-s − 0.541·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 + 3.41T + 8T^{2} \) |
| 3 | \( 1 + 3.08T + 27T^{2} \) |
| 7 | \( 1 + 22.1T + 343T^{2} \) |
| 11 | \( 1 + 32.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 26.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 73.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 81.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 217.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 60.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 79.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 9.73T + 7.95e4T^{2} \) |
| 47 | \( 1 + 343.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 270.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 647.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 600.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 593.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 873.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 310.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 76.4T + 9.12e5T^{2} \) |
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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
−8.222119220157469263321798575004, −8.055414631293184299351823523750, −6.74838435808082044942278815192, −6.40989116093391952573331869577, −5.22868625923694578765269188922, −4.39993460231698346062908786106, −3.17347017234284833265077137618, −2.21595677956214797159895817966, −0.61548212497450689319667100815, 0,
0.61548212497450689319667100815, 2.21595677956214797159895817966, 3.17347017234284833265077137618, 4.39993460231698346062908786106, 5.22868625923694578765269188922, 6.40989116093391952573331869577, 6.74838435808082044942278815192, 8.055414631293184299351823523750, 8.222119220157469263321798575004
