| L(s) = 1 | − 2.69·2-s + 3-s + 5.28·4-s + 2.43·5-s − 2.69·6-s − 1.23·7-s − 8.85·8-s + 9-s − 6.58·10-s + 3.49·11-s + 5.28·12-s − 3.44·13-s + 3.33·14-s + 2.43·15-s + 13.3·16-s + 17-s − 2.69·18-s + 1.20·19-s + 12.8·20-s − 1.23·21-s − 9.42·22-s − 4.65·23-s − 8.85·24-s + 0.946·25-s + 9.28·26-s + 27-s − 6.51·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.577·3-s + 2.64·4-s + 1.09·5-s − 1.10·6-s − 0.466·7-s − 3.13·8-s + 0.333·9-s − 2.08·10-s + 1.05·11-s + 1.52·12-s − 0.954·13-s + 0.890·14-s + 0.629·15-s + 3.33·16-s + 0.242·17-s − 0.636·18-s + 0.277·19-s + 2.87·20-s − 0.269·21-s − 2.00·22-s − 0.970·23-s − 1.80·24-s + 0.189·25-s + 1.82·26-s + 0.192·27-s − 1.23·28-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 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 - 0.712T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 + 0.770T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 8.40T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 67 | \( 1 - 3.74T + 67T^{2} \) |
| 71 | \( 1 + 3.98T + 71T^{2} \) |
| 73 | \( 1 - 2.71T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 0.952T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
| 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
−7.78917996692549089680960653986, −6.85935368610379820146027330889, −6.48103365325596244184421142193, −5.87323374610109945655346553100, −4.73127729272889090804089579882, −3.39350247398545976648548234891, −2.74991123219838366728455314445, −1.85575960180934742132046576736, −1.38675932714772188472937900348, 0,
1.38675932714772188472937900348, 1.85575960180934742132046576736, 2.74991123219838366728455314445, 3.39350247398545976648548234891, 4.73127729272889090804089579882, 5.87323374610109945655346553100, 6.48103365325596244184421142193, 6.85935368610379820146027330889, 7.78917996692549089680960653986
