| L(s) = 1 | − 1.32·2-s + 3-s − 0.236·4-s + 2.41·5-s − 1.32·6-s + 0.553·7-s + 2.97·8-s + 9-s − 3.20·10-s − 0.0504·11-s − 0.236·12-s − 2.00·13-s − 0.735·14-s + 2.41·15-s − 3.47·16-s + 17-s − 1.32·18-s − 7.25·19-s − 0.571·20-s + 0.553·21-s + 0.0669·22-s + 3.59·23-s + 2.97·24-s + 0.826·25-s + 2.66·26-s + 27-s − 0.131·28-s + ⋯ |
| L(s) = 1 | − 0.938·2-s + 0.577·3-s − 0.118·4-s + 1.07·5-s − 0.542·6-s + 0.209·7-s + 1.05·8-s + 0.333·9-s − 1.01·10-s − 0.0152·11-s − 0.0683·12-s − 0.557·13-s − 0.196·14-s + 0.623·15-s − 0.867·16-s + 0.242·17-s − 0.312·18-s − 1.66·19-s − 0.127·20-s + 0.120·21-s + 0.0142·22-s + 0.749·23-s + 0.606·24-s + 0.165·25-s + 0.523·26-s + 0.192·27-s − 0.0247·28-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 | \( 1 + 1.32T + 2T^{2} \) |
| 5 | \( 1 - 2.41T + 5T^{2} \) |
| 7 | \( 1 - 0.553T + 7T^{2} \) |
| 11 | \( 1 + 0.0504T + 11T^{2} \) |
| 13 | \( 1 + 2.00T + 13T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 + 1.39T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 3.35T + 43T^{2} \) |
| 47 | \( 1 - 2.01T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 6.27T + 59T^{2} \) |
| 61 | \( 1 + 4.50T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 9.22T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 + 4.22T + 79T^{2} \) |
| 83 | \( 1 - 7.35T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
| 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
−7.55585089575667103996015492682, −7.09125793388499491505339696908, −6.19945551981178041051408776486, −5.41196427734092927049041507486, −4.64624521484510115254140384567, −3.96085978764623690742366329726, −2.78790687285660126397899509272, −1.99907723232416505388158224398, −1.38686061500943257456984130914, 0,
1.38686061500943257456984130914, 1.99907723232416505388158224398, 2.78790687285660126397899509272, 3.96085978764623690742366329726, 4.64624521484510115254140384567, 5.41196427734092927049041507486, 6.19945551981178041051408776486, 7.09125793388499491505339696908, 7.55585089575667103996015492682
