| L(s) = 1 | + 2-s − 0.158·3-s + 4-s + 1.97·5-s − 0.158·6-s + 1.09·7-s + 8-s − 2.97·9-s + 1.97·10-s − 1.52·11-s − 0.158·12-s − 2.04·13-s + 1.09·14-s − 0.313·15-s + 16-s − 4.89·17-s − 2.97·18-s − 3.72·19-s + 1.97·20-s − 0.173·21-s − 1.52·22-s + 5.32·23-s − 0.158·24-s − 1.10·25-s − 2.04·26-s + 0.948·27-s + 1.09·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0916·3-s + 0.5·4-s + 0.882·5-s − 0.0647·6-s + 0.414·7-s + 0.353·8-s − 0.991·9-s + 0.624·10-s − 0.458·11-s − 0.0458·12-s − 0.566·13-s + 0.292·14-s − 0.0808·15-s + 0.250·16-s − 1.18·17-s − 0.701·18-s − 0.853·19-s + 0.441·20-s − 0.0379·21-s − 0.324·22-s + 1.11·23-s − 0.0323·24-s − 0.221·25-s − 0.400·26-s + 0.182·27-s + 0.207·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 0.158T + 3T^{2} \) |
| 5 | \( 1 - 1.97T + 5T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 11 | \( 1 + 1.52T + 11T^{2} \) |
| 13 | \( 1 + 2.04T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 - 1.10T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 - 3.11T + 41T^{2} \) |
| 43 | \( 1 + 7.96T + 43T^{2} \) |
| 47 | \( 1 + 7.37T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 + 0.229T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 - 2.33T + 71T^{2} \) |
| 73 | \( 1 - 7.58T + 73T^{2} \) |
| 79 | \( 1 - 8.51T + 79T^{2} \) |
| 83 | \( 1 + 7.62T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
| 97 | \( 1 + 6.53T + 97T^{2} \) |
| 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
−7.73337465441855877911959380549, −6.66329573824934793212113010498, −6.34177791872757054108987733770, −5.36635008906913853889986627351, −5.05621729208215108943245825065, −4.20243268048414361503754319049, −3.08749686307994934178251200214, −2.39860682874112408085418792837, −1.70208579952708774924837789317, 0,
1.70208579952708774924837789317, 2.39860682874112408085418792837, 3.08749686307994934178251200214, 4.20243268048414361503754319049, 5.05621729208215108943245825065, 5.36635008906913853889986627351, 6.34177791872757054108987733770, 6.66329573824934793212113010498, 7.73337465441855877911959380549
