| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·5-s − 2·6-s + 7-s + 8-s + 9-s − 2·10-s − 4·11-s − 2·12-s + 2·13-s + 14-s + 4·15-s + 16-s + 18-s − 2·19-s − 2·20-s − 2·21-s − 4·22-s − 23-s − 2·24-s − 25-s + 2·26-s + 4·27-s + 28-s + 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.577·12-s + 0.554·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.447·20-s − 0.436·21-s − 0.852·22-s − 0.208·23-s − 0.408·24-s − 1/5·25-s + 0.392·26-s + 0.769·27-s + 0.188·28-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.02076658421919, −12.39494827634074, −12.12943602005689, −11.57394907143715, −11.25984146677067, −10.83592132573387, −10.65312000913336, −9.943951060687235, −9.467913543944251, −8.589975498159707, −8.144368911681048, −7.961653630213916, −7.235840723983273, −6.794279482285717, −6.330965459306657, −5.739138309073427, −5.317104411779455, −4.991558930639943, −4.464079366864442, −3.870182429342392, −3.398801132986382, −2.842642724209554, −2.012440881271181, −1.539283050996870, −0.5333769338030790, 0,
0.5333769338030790, 1.539283050996870, 2.012440881271181, 2.842642724209554, 3.398801132986382, 3.870182429342392, 4.464079366864442, 4.991558930639943, 5.317104411779455, 5.739138309073427, 6.330965459306657, 6.794279482285717, 7.235840723983273, 7.961653630213916, 8.144368911681048, 8.589975498159707, 9.467913543944251, 9.943951060687235, 10.65312000913336, 10.83592132573387, 11.25984146677067, 11.57394907143715, 12.12943602005689, 12.39494827634074, 13.02076658421919
