L(s) = 1 | + 2-s + 1.58·3-s + 4-s + 3.09·5-s + 1.58·6-s − 0.0617·7-s + 8-s − 0.486·9-s + 3.09·10-s + 3.30·11-s + 1.58·12-s + 3.79·13-s − 0.0617·14-s + 4.90·15-s + 16-s − 6.03·17-s − 0.486·18-s + 1.36·19-s + 3.09·20-s − 0.0978·21-s + 3.30·22-s − 23-s + 1.58·24-s + 4.56·25-s + 3.79·26-s − 5.52·27-s − 0.0617·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.915·3-s + 0.5·4-s + 1.38·5-s + 0.647·6-s − 0.0233·7-s + 0.353·8-s − 0.162·9-s + 0.978·10-s + 0.997·11-s + 0.457·12-s + 1.05·13-s − 0.0164·14-s + 1.26·15-s + 0.250·16-s − 1.46·17-s − 0.114·18-s + 0.312·19-s + 0.691·20-s − 0.0213·21-s + 0.705·22-s − 0.208·23-s + 0.323·24-s + 0.913·25-s + 0.744·26-s − 1.06·27-s − 0.0116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.237365601\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.237365601\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 + 0.0617T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 0.0755T + 37T^{2} \) |
| 41 | \( 1 - 6.19T + 41T^{2} \) |
| 43 | \( 1 - 5.45T + 43T^{2} \) |
| 47 | \( 1 - 8.47T + 47T^{2} \) |
| 53 | \( 1 - 9.59T + 53T^{2} \) |
| 59 | \( 1 + 5.33T + 59T^{2} \) |
| 61 | \( 1 + 0.474T + 61T^{2} \) |
| 67 | \( 1 - 5.51T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 + 7.27T + 83T^{2} \) |
| 89 | \( 1 + 6.67T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.176603435296453566917162122392, −7.23271145714584548026356109620, −6.37979192979216969569514121064, −6.06337447970491939409144792523, −5.30713049111299526672126487247, −4.23430124630144379773440763459, −3.69615706986440488877600895563, −2.66340129206742309884956488925, −2.15160400498673403376919328832, −1.24771390412918518250981637755,
1.24771390412918518250981637755, 2.15160400498673403376919328832, 2.66340129206742309884956488925, 3.69615706986440488877600895563, 4.23430124630144379773440763459, 5.30713049111299526672126487247, 6.06337447970491939409144792523, 6.37979192979216969569514121064, 7.23271145714584548026356109620, 8.176603435296453566917162122392