| L(s) = 1 | + 0.871·2-s − 2.06·3-s − 1.24·4-s + 1.68·5-s − 1.79·6-s − 2.13·7-s − 2.82·8-s + 1.24·9-s + 1.46·10-s + 5.16·11-s + 2.55·12-s + 2.30·13-s − 1.86·14-s − 3.46·15-s + 0.0196·16-s − 0.452·17-s + 1.08·18-s − 0.874·19-s − 2.08·20-s + 4.39·21-s + 4.49·22-s + 1.75·23-s + 5.81·24-s − 2.16·25-s + 2.00·26-s + 3.61·27-s + 2.64·28-s + ⋯ |
| L(s) = 1 | + 0.616·2-s − 1.18·3-s − 0.620·4-s + 0.752·5-s − 0.733·6-s − 0.806·7-s − 0.998·8-s + 0.415·9-s + 0.463·10-s + 1.55·11-s + 0.737·12-s + 0.639·13-s − 0.497·14-s − 0.895·15-s + 0.00491·16-s − 0.109·17-s + 0.255·18-s − 0.200·19-s − 0.466·20-s + 0.960·21-s + 0.959·22-s + 0.365·23-s + 1.18·24-s − 0.433·25-s + 0.394·26-s + 0.695·27-s + 0.500·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1511 | \( 1 + T \) |
| good | 2 | \( 1 - 0.871T + 2T^{2} \) |
| 3 | \( 1 + 2.06T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 0.452T + 17T^{2} \) |
| 19 | \( 1 + 0.874T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 0.776T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 4.21T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 8.40T + 47T^{2} \) |
| 53 | \( 1 - 6.37T + 53T^{2} \) |
| 59 | \( 1 + 5.33T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 - 2.42T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 0.520T + 89T^{2} \) |
| 97 | \( 1 - 4.54T + 97T^{2} \) |
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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
−9.222445476949383232632147334723, −8.560256911116892772549359856441, −6.94556416062839003910632123568, −6.29968435802647771583650434233, −5.82068482865077542350253184181, −5.07704372211990489053761327070, −4.02675812946286497220533758300, −3.28979803070580147208822931839, −1.52127053328778139496796634154, 0,
1.52127053328778139496796634154, 3.28979803070580147208822931839, 4.02675812946286497220533758300, 5.07704372211990489053761327070, 5.82068482865077542350253184181, 6.29968435802647771583650434233, 6.94556416062839003910632123568, 8.560256911116892772549359856441, 9.222445476949383232632147334723
