| L(s) = 1 | + 1.61·2-s − 3-s + 0.621·4-s + 2.18·5-s − 1.61·6-s − 1.71·7-s − 2.23·8-s + 9-s + 3.53·10-s − 0.0394·11-s − 0.621·12-s − 1.76·13-s − 2.77·14-s − 2.18·15-s − 4.85·16-s + 17-s + 1.61·18-s + 2.40·19-s + 1.35·20-s + 1.71·21-s − 0.0638·22-s + 6.02·23-s + 2.23·24-s − 0.238·25-s − 2.85·26-s − 27-s − 1.06·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.310·4-s + 0.975·5-s − 0.660·6-s − 0.647·7-s − 0.789·8-s + 0.333·9-s + 1.11·10-s − 0.0118·11-s − 0.179·12-s − 0.488·13-s − 0.741·14-s − 0.563·15-s − 1.21·16-s + 0.242·17-s + 0.381·18-s + 0.552·19-s + 0.303·20-s + 0.374·21-s − 0.0136·22-s + 1.25·23-s + 0.455·24-s − 0.0477·25-s − 0.559·26-s − 0.192·27-s − 0.201·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 + 0.0394T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 19 | \( 1 - 2.40T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 + 1.83T + 29T^{2} \) |
| 31 | \( 1 + 4.03T + 31T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 + 4.88T + 43T^{2} \) |
| 47 | \( 1 + 4.46T + 47T^{2} \) |
| 53 | \( 1 - 0.363T + 53T^{2} \) |
| 59 | \( 1 - 4.60T + 59T^{2} \) |
| 61 | \( 1 + 6.04T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 0.718T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
| 97 | \( 1 + 8.58T + 97T^{2} \) |
| 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
−7.06050216189789817428826155935, −6.59974101622754132093835202082, −5.84545426585994582219808064446, −5.42759632591497284722209370989, −4.85361093448139271268447932064, −4.01599639144090691057905930417, −3.16832648858131172492533968719, −2.53359385010658808073430430490, −1.37258676727091824639766766897, 0,
1.37258676727091824639766766897, 2.53359385010658808073430430490, 3.16832648858131172492533968719, 4.01599639144090691057905930417, 4.85361093448139271268447932064, 5.42759632591497284722209370989, 5.84545426585994582219808064446, 6.59974101622754132093835202082, 7.06050216189789817428826155935
