| L(s) = 1 | − 2-s + 1.61·3-s + 4-s + 2.23·5-s − 1.61·6-s + 2·7-s − 8-s − 0.381·9-s − 2.23·10-s + 0.618·11-s + 1.61·12-s − 3·13-s − 2·14-s + 3.61·15-s + 16-s − 3·17-s + 0.381·18-s − 19-s + 2.23·20-s + 3.23·21-s − 0.618·22-s + 3·23-s − 1.61·24-s + 3·26-s − 5.47·27-s + 2·28-s − 6.85·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.934·3-s + 0.5·4-s + 0.999·5-s − 0.660·6-s + 0.755·7-s − 0.353·8-s − 0.127·9-s − 0.707·10-s + 0.186·11-s + 0.467·12-s − 0.832·13-s − 0.534·14-s + 0.934·15-s + 0.250·16-s − 0.727·17-s + 0.0900·18-s − 0.229·19-s + 0.499·20-s + 0.706·21-s − 0.131·22-s + 0.625·23-s − 0.330·24-s + 0.588·26-s − 1.05·27-s + 0.377·28-s − 1.27·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 6.85T + 29T^{2} \) |
| 31 | \( 1 - 2.61T + 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 + 9.18T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 7.47T + 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 0.291T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 0.236T + 97T^{2} \) |
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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
−7.74305080200683589186518069485, −6.87774095289158745493748963022, −6.33089491160257623395421785306, −5.32356618169937814556118154033, −4.85015858437178472902272334762, −3.63313256278593732921444578019, −2.86861698525612983896027321234, −1.92773251985947286974626800766, −1.73392881987348740330249936915, 0,
1.73392881987348740330249936915, 1.92773251985947286974626800766, 2.86861698525612983896027321234, 3.63313256278593732921444578019, 4.85015858437178472902272334762, 5.32356618169937814556118154033, 6.33089491160257623395421785306, 6.87774095289158745493748963022, 7.74305080200683589186518069485
