| L(s) = 1 | + 1.04·2-s + 3-s − 0.915·4-s − 1.30·5-s + 1.04·6-s + 3.53·7-s − 3.03·8-s + 9-s − 1.35·10-s + 3.63·11-s − 0.915·12-s + 2.12·13-s + 3.67·14-s − 1.30·15-s − 1.33·16-s − 17-s + 1.04·18-s − 6.54·19-s + 1.19·20-s + 3.53·21-s + 3.78·22-s − 9.46·23-s − 3.03·24-s − 3.30·25-s + 2.21·26-s + 27-s − 3.23·28-s + ⋯ |
| L(s) = 1 | + 0.736·2-s + 0.577·3-s − 0.457·4-s − 0.582·5-s + 0.425·6-s + 1.33·7-s − 1.07·8-s + 0.333·9-s − 0.428·10-s + 1.09·11-s − 0.264·12-s + 0.589·13-s + 0.983·14-s − 0.336·15-s − 0.332·16-s − 0.242·17-s + 0.245·18-s − 1.50·19-s + 0.266·20-s + 0.770·21-s + 0.806·22-s − 1.97·23-s − 0.619·24-s − 0.660·25-s + 0.434·26-s + 0.192·27-s − 0.611·28-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 | \( 1 - 1.04T + 2T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 11 | \( 1 - 3.63T + 11T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 19 | \( 1 + 6.54T + 19T^{2} \) |
| 23 | \( 1 + 9.46T + 23T^{2} \) |
| 29 | \( 1 + 5.38T + 29T^{2} \) |
| 31 | \( 1 - 3.92T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 + 9.21T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 3.29T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 61 | \( 1 + 0.736T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 7.08T + 71T^{2} \) |
| 73 | \( 1 - 8.51T + 73T^{2} \) |
| 79 | \( 1 - 5.94T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 5.14T + 89T^{2} \) |
| 97 | \( 1 + 1.15T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84823673452257987107138499041, −6.46540134727024646805790006913, −6.25916272941436834047527318817, −5.10043645021134611103639793461, −4.51009542966259822729447899595, −3.88117185131382699290759744461, −3.59012537694997671072339124951, −2.20338594838448561386920777680, −1.53451937238242466821858401346, 0,
1.53451937238242466821858401346, 2.20338594838448561386920777680, 3.59012537694997671072339124951, 3.88117185131382699290759744461, 4.51009542966259822729447899595, 5.10043645021134611103639793461, 6.25916272941436834047527318817, 6.46540134727024646805790006913, 7.84823673452257987107138499041
