| L(s) = 1 | − 0.524·2-s + 3-s − 1.72·4-s − 0.0309·5-s − 0.524·6-s + 4.53·7-s + 1.95·8-s + 9-s + 0.0162·10-s − 5.92·11-s − 1.72·12-s + 1.49·13-s − 2.37·14-s − 0.0309·15-s + 2.42·16-s + 17-s − 0.524·18-s − 0.562·19-s + 0.0533·20-s + 4.53·21-s + 3.11·22-s + 2.44·23-s + 1.95·24-s − 4.99·25-s − 0.783·26-s + 27-s − 7.81·28-s + ⋯ |
| L(s) = 1 | − 0.371·2-s + 0.577·3-s − 0.862·4-s − 0.0138·5-s − 0.214·6-s + 1.71·7-s + 0.691·8-s + 0.333·9-s + 0.00513·10-s − 1.78·11-s − 0.497·12-s + 0.413·13-s − 0.635·14-s − 0.00798·15-s + 0.605·16-s + 0.242·17-s − 0.123·18-s − 0.129·19-s + 0.0119·20-s + 0.988·21-s + 0.663·22-s + 0.509·23-s + 0.399·24-s − 0.999·25-s − 0.153·26-s + 0.192·27-s − 1.47·28-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 | \( 1 + 0.524T + 2T^{2} \) |
| 5 | \( 1 + 0.0309T + 5T^{2} \) |
| 7 | \( 1 - 4.53T + 7T^{2} \) |
| 11 | \( 1 + 5.92T + 11T^{2} \) |
| 13 | \( 1 - 1.49T + 13T^{2} \) |
| 19 | \( 1 + 0.562T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 + 0.878T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 7.87T + 47T^{2} \) |
| 53 | \( 1 - 0.351T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 + 0.530T + 61T^{2} \) |
| 67 | \( 1 - 1.53T + 67T^{2} \) |
| 71 | \( 1 - 1.53T + 71T^{2} \) |
| 73 | \( 1 + 6.42T + 73T^{2} \) |
| 79 | \( 1 + 0.447T + 79T^{2} \) |
| 83 | \( 1 - 4.15T + 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.73350331947965982974016766432, −7.28131443295212497301735528529, −5.88065420861148334527864883648, −5.16813816001965347968188152682, −4.78291496795450578999597730065, −3.99325722485182541268713412540, −3.08257114092282103685860603163, −2.03621673639413230021130519613, −1.35833973890169431177745663156, 0,
1.35833973890169431177745663156, 2.03621673639413230021130519613, 3.08257114092282103685860603163, 3.99325722485182541268713412540, 4.78291496795450578999597730065, 5.16813816001965347968188152682, 5.88065420861148334527864883648, 7.28131443295212497301735528529, 7.73350331947965982974016766432
