| L(s) = 1 | + 1.80·2-s − 0.875·3-s + 1.24·4-s + 5-s − 1.57·6-s + 0.602·7-s − 1.35·8-s − 2.23·9-s + 1.80·10-s + 2.35·11-s − 1.09·12-s − 3.77·13-s + 1.08·14-s − 0.875·15-s − 4.93·16-s − 2.66·17-s − 4.02·18-s − 1.96·19-s + 1.24·20-s − 0.527·21-s + 4.24·22-s − 8.16·23-s + 1.18·24-s + 25-s − 6.80·26-s + 4.58·27-s + 0.752·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 0.505·3-s + 0.624·4-s + 0.447·5-s − 0.644·6-s + 0.227·7-s − 0.478·8-s − 0.744·9-s + 0.569·10-s + 0.709·11-s − 0.315·12-s − 1.04·13-s + 0.290·14-s − 0.226·15-s − 1.23·16-s − 0.647·17-s − 0.948·18-s − 0.451·19-s + 0.279·20-s − 0.115·21-s + 0.904·22-s − 1.70·23-s + 0.242·24-s + 0.200·25-s − 1.33·26-s + 0.881·27-s + 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 3 | \( 1 + 0.875T + 3T^{2} \) |
| 7 | \( 1 - 0.602T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 23 | \( 1 + 8.16T + 23T^{2} \) |
| 29 | \( 1 - 3.36T + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 - 0.597T + 37T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 + 1.17T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 0.0668T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 - 4.09T + 79T^{2} \) |
| 83 | \( 1 + 4.02T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
| 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
−8.931157185169463613806343651354, −7.85147679869138662384816735438, −6.79243931418184493369828771007, −6.05634030870514973637996166389, −5.60613477781826714559582794390, −4.66374791222623339361931935255, −4.08218654306292264277512066781, −2.89342961030175570073879950527, −2.00668417381473655817254847921, 0,
2.00668417381473655817254847921, 2.89342961030175570073879950527, 4.08218654306292264277512066781, 4.66374791222623339361931935255, 5.60613477781826714559582794390, 6.05634030870514973637996166389, 6.79243931418184493369828771007, 7.85147679869138662384816735438, 8.931157185169463613806343651354
