| L(s) = 1 | − 1.41·2-s + 3-s + 0.00726·4-s + 2.31·5-s − 1.41·6-s + 2.67·7-s + 2.82·8-s + 9-s − 3.27·10-s − 4.03·11-s + 0.00726·12-s − 3.81·13-s − 3.79·14-s + 2.31·15-s − 4.01·16-s − 17-s − 1.41·18-s − 5.43·19-s + 0.0167·20-s + 2.67·21-s + 5.71·22-s − 4.96·23-s + 2.82·24-s + 0.339·25-s + 5.40·26-s + 27-s + 0.0194·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.577·3-s + 0.00363·4-s + 1.03·5-s − 0.578·6-s + 1.01·7-s + 0.998·8-s + 0.333·9-s − 1.03·10-s − 1.21·11-s + 0.00209·12-s − 1.05·13-s − 1.01·14-s + 0.596·15-s − 1.00·16-s − 0.242·17-s − 0.333·18-s − 1.24·19-s + 0.00375·20-s + 0.584·21-s + 1.21·22-s − 1.03·23-s + 0.576·24-s + 0.0678·25-s + 1.06·26-s + 0.192·27-s + 0.00367·28-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 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 - 2.31T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 + 3.81T + 13T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 + 4.96T + 23T^{2} \) |
| 29 | \( 1 - 7.33T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 37 | \( 1 + 4.21T + 37T^{2} \) |
| 41 | \( 1 + 0.767T + 41T^{2} \) |
| 43 | \( 1 + 0.0132T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 - 7.15T + 53T^{2} \) |
| 59 | \( 1 + 1.07T + 59T^{2} \) |
| 61 | \( 1 - 5.20T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 8.38T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 5.99T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.889403477154054819798872955416, −7.01400209575136821539316050876, −6.27554688004160904226124112234, −5.16351780092602078947359077621, −4.83714546598012313014539627980, −4.02635990947359875259275830017, −2.45287375665350978622690069550, −2.28691419607285577966691080768, −1.32787373782058694017965704521, 0,
1.32787373782058694017965704521, 2.28691419607285577966691080768, 2.45287375665350978622690069550, 4.02635990947359875259275830017, 4.83714546598012313014539627980, 5.16351780092602078947359077621, 6.27554688004160904226124112234, 7.01400209575136821539316050876, 7.889403477154054819798872955416
