L(s) = 1 | − 1.41·2-s − 0.517·3-s + 0.732·6-s + 2.44·7-s + 2.82·8-s − 2.73·9-s − 4.73·11-s + 2.44·13-s − 3.46·14-s − 4.00·16-s − 2.96·17-s + 3.86·18-s + 3.19·19-s − 1.26·21-s + 6.69·22-s + 1.41·23-s − 1.46·24-s − 3.46·26-s + 2.96·27-s + 2.19·29-s + 31-s + 2.44·33-s + 4.19·34-s + 0.896·37-s + ⋯ |
L(s) = 1 | − 1.00·2-s − 0.298·3-s + 0.298·6-s + 0.925·7-s + 0.999·8-s − 0.910·9-s − 1.42·11-s + 0.679·13-s − 0.925·14-s − 1.00·16-s − 0.719·17-s + 0.910·18-s + 0.733·19-s − 0.276·21-s + 1.42·22-s + 0.294·23-s − 0.298·24-s − 0.679·26-s + 0.571·27-s + 0.407·29-s + 0.179·31-s + 0.426·33-s + 0.719·34-s + 0.147·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 + 0.517T + 3T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 37 | \( 1 - 0.896T + 37T^{2} \) |
| 41 | \( 1 + 5.53T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 8.76T + 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−9.892600906458298267112480021863, −8.866937316893820636768531300804, −8.229158007710173622805720354778, −7.73330849797182257237324414555, −6.49724580851037139839715935277, −5.24376718538276076646815071170, −4.72726999262180620280217128031, −3.05366264353249877700281613486, −1.60341138575478556647579058276, 0,
1.60341138575478556647579058276, 3.05366264353249877700281613486, 4.72726999262180620280217128031, 5.24376718538276076646815071170, 6.49724580851037139839715935277, 7.73330849797182257237324414555, 8.229158007710173622805720354778, 8.866937316893820636768531300804, 9.892600906458298267112480021863