| L(s) = 1 | − 0.857·2-s − 3-s − 1.26·4-s + 0.241·5-s + 0.857·6-s − 3.40·7-s + 2.79·8-s + 9-s − 0.207·10-s + 4.52·11-s + 1.26·12-s − 3.66·13-s + 2.92·14-s − 0.241·15-s + 0.130·16-s − 17-s − 0.857·18-s + 3.02·19-s − 0.305·20-s + 3.40·21-s − 3.87·22-s − 6.34·23-s − 2.79·24-s − 4.94·25-s + 3.14·26-s − 27-s + 4.31·28-s + ⋯ |
| L(s) = 1 | − 0.606·2-s − 0.577·3-s − 0.632·4-s + 0.108·5-s + 0.349·6-s − 1.28·7-s + 0.989·8-s + 0.333·9-s − 0.0655·10-s + 1.36·11-s + 0.365·12-s − 1.01·13-s + 0.781·14-s − 0.0624·15-s + 0.0327·16-s − 0.242·17-s − 0.202·18-s + 0.693·19-s − 0.0684·20-s + 0.743·21-s − 0.827·22-s − 1.32·23-s − 0.571·24-s − 0.988·25-s + 0.616·26-s − 0.192·27-s + 0.815·28-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 | \( 1 + 0.857T + 2T^{2} \) |
| 5 | \( 1 - 0.241T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 19 | \( 1 - 3.02T + 19T^{2} \) |
| 23 | \( 1 + 6.34T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 - 2.05T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 - 0.671T + 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 + 0.151T + 47T^{2} \) |
| 53 | \( 1 + 8.36T + 53T^{2} \) |
| 59 | \( 1 - 1.97T + 59T^{2} \) |
| 61 | \( 1 - 0.358T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 + 4.67T + 71T^{2} \) |
| 73 | \( 1 - 0.743T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 4.57T + 83T^{2} \) |
| 89 | \( 1 - 1.89T + 89T^{2} \) |
| 97 | \( 1 + 1.97T + 97T^{2} \) |
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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
−7.59022500808131176005093506865, −6.63400245831892718473920237417, −6.32993506075207654060025766127, −5.47562055374689058783825836277, −4.56771285376727127094059095938, −4.03227713477740032442800553465, −3.19373442266329157372028994520, −2.00397163400179779676860573235, −0.914792416765416847664497004550, 0,
0.914792416765416847664497004550, 2.00397163400179779676860573235, 3.19373442266329157372028994520, 4.03227713477740032442800553465, 4.56771285376727127094059095938, 5.47562055374689058783825836277, 6.32993506075207654060025766127, 6.63400245831892718473920237417, 7.59022500808131176005093506865
