| L(s) = 1 | + 1.82·2-s − 2·3-s − 4.65·4-s − 10.4·5-s − 3.65·6-s + 7·7-s − 23.1·8-s − 23·9-s − 19.1·10-s − 11·11-s + 9.31·12-s − 34.3·13-s + 12.7·14-s + 20.9·15-s − 5.05·16-s + 111.·17-s − 42.0·18-s − 56.4·19-s + 48.8·20-s − 14·21-s − 20.1·22-s + 22.9·23-s + 46.2·24-s − 15.0·25-s − 62.7·26-s + 100·27-s − 32.5·28-s + ⋯ |
| L(s) = 1 | + 0.646·2-s − 0.384·3-s − 0.582·4-s − 0.937·5-s − 0.248·6-s + 0.377·7-s − 1.02·8-s − 0.851·9-s − 0.606·10-s − 0.301·11-s + 0.224·12-s − 0.732·13-s + 0.244·14-s + 0.360·15-s − 0.0790·16-s + 1.59·17-s − 0.550·18-s − 0.682·19-s + 0.545·20-s − 0.145·21-s − 0.194·22-s + 0.207·23-s + 0.393·24-s − 0.120·25-s − 0.473·26-s + 0.712·27-s − 0.220·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 1.82T + 8T^{2} \) |
| 3 | \( 1 + 2T + 27T^{2} \) |
| 5 | \( 1 + 10.4T + 125T^{2} \) |
| 13 | \( 1 + 34.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 56.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 252.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 230.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 316T + 7.95e4T^{2} \) |
| 47 | \( 1 + 27.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 502.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 433.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 802.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 944.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 405.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 936.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 433.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−13.46182604614190090649654697807, −12.16636564393552817015413289321, −11.70321035403359102417093658095, −10.21663800740245219000175387073, −8.699425550353964800658599880631, −7.65144753574054899734687000460, −5.79346959615263852936035718431, −4.74642547211026159586313813258, −3.31421134268810902426488742803, 0,
3.31421134268810902426488742803, 4.74642547211026159586313813258, 5.79346959615263852936035718431, 7.65144753574054899734687000460, 8.699425550353964800658599880631, 10.21663800740245219000175387073, 11.70321035403359102417093658095, 12.16636564393552817015413289321, 13.46182604614190090649654697807
