| L(s) = 1 | + 1.20·2-s + 3-s − 0.557·4-s − 4.37·5-s + 1.20·6-s − 0.896·7-s − 3.07·8-s + 9-s − 5.25·10-s + 0.0156·11-s − 0.557·12-s + 4.65·13-s − 1.07·14-s − 4.37·15-s − 2.57·16-s − 17-s + 1.20·18-s − 0.462·19-s + 2.43·20-s − 0.896·21-s + 0.0187·22-s + 3.14·23-s − 3.07·24-s + 14.1·25-s + 5.59·26-s + 27-s + 0.499·28-s + ⋯ |
| L(s) = 1 | + 0.849·2-s + 0.577·3-s − 0.278·4-s − 1.95·5-s + 0.490·6-s − 0.338·7-s − 1.08·8-s + 0.333·9-s − 1.66·10-s + 0.00471·11-s − 0.160·12-s + 1.29·13-s − 0.287·14-s − 1.12·15-s − 0.643·16-s − 0.242·17-s + 0.283·18-s − 0.106·19-s + 0.544·20-s − 0.195·21-s + 0.00400·22-s + 0.656·23-s − 0.626·24-s + 2.82·25-s + 1.09·26-s + 0.192·27-s + 0.0943·28-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 | \( 1 - 1.20T + 2T^{2} \) |
| 5 | \( 1 + 4.37T + 5T^{2} \) |
| 7 | \( 1 + 0.896T + 7T^{2} \) |
| 11 | \( 1 - 0.0156T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 19 | \( 1 + 0.462T + 19T^{2} \) |
| 23 | \( 1 - 3.14T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 + 2.56T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 - 4.83T + 43T^{2} \) |
| 47 | \( 1 + 6.79T + 47T^{2} \) |
| 53 | \( 1 - 2.50T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 + 8.90T + 71T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 + 7.29T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 9.59T + 97T^{2} \) |
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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
−7.63170443242083398776458253917, −6.65655093514046675845949120412, −6.26550208922882941529307682286, −4.95931802294642369512775125330, −4.58621934886310537664092983120, −3.76923830844477246670934490310, −3.40562253207430301330463610214, −2.81124601592579582424736361702, −1.12832845045148973975759589311, 0,
1.12832845045148973975759589311, 2.81124601592579582424736361702, 3.40562253207430301330463610214, 3.76923830844477246670934490310, 4.58621934886310537664092983120, 4.95931802294642369512775125330, 6.26550208922882941529307682286, 6.65655093514046675845949120412, 7.63170443242083398776458253917
