| L(s) = 1 | − 1.97·2-s − 3-s + 1.89·4-s + 1.34·5-s + 1.97·6-s − 4.50·7-s + 0.198·8-s + 9-s − 2.65·10-s − 0.344·11-s − 1.89·12-s + 0.515·13-s + 8.89·14-s − 1.34·15-s − 4.19·16-s + 17-s − 1.97·18-s + 6.63·19-s + 2.55·20-s + 4.50·21-s + 0.679·22-s − 2.14·23-s − 0.198·24-s − 3.19·25-s − 1.01·26-s − 27-s − 8.55·28-s + ⋯ |
| L(s) = 1 | − 1.39·2-s − 0.577·3-s + 0.949·4-s + 0.601·5-s + 0.806·6-s − 1.70·7-s + 0.0702·8-s + 0.333·9-s − 0.839·10-s − 0.103·11-s − 0.548·12-s + 0.142·13-s + 2.37·14-s − 0.347·15-s − 1.04·16-s + 0.242·17-s − 0.465·18-s + 1.52·19-s + 0.571·20-s + 0.983·21-s + 0.144·22-s − 0.447·23-s − 0.0405·24-s − 0.638·25-s − 0.199·26-s − 0.192·27-s − 1.61·28-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 | \( 1 + 1.97T + 2T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 4.50T + 7T^{2} \) |
| 11 | \( 1 + 0.344T + 11T^{2} \) |
| 13 | \( 1 - 0.515T + 13T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 - 3.34T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 + 8.26T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 - 1.22T + 61T^{2} \) |
| 67 | \( 1 - 2.38T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 5.95T + 73T^{2} \) |
| 79 | \( 1 + 0.348T + 79T^{2} \) |
| 83 | \( 1 + 1.98T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−7.54864406141604458454762376663, −6.82247936478338444464458082217, −6.31296067243497484939705398772, −5.68473104274776763205105847723, −4.83600697732321139885519762680, −3.70084958225182474804966424932, −2.94972089579907759302516535611, −1.90550718234436861090117091936, −0.927969327819950008168249786073, 0,
0.927969327819950008168249786073, 1.90550718234436861090117091936, 2.94972089579907759302516535611, 3.70084958225182474804966424932, 4.83600697732321139885519762680, 5.68473104274776763205105847723, 6.31296067243497484939705398772, 6.82247936478338444464458082217, 7.54864406141604458454762376663
