L(s) = 1 | + 0.870·2-s − 1.24·4-s − 3.50·5-s − 4.57·7-s − 2.82·8-s − 3.04·10-s + 11-s + 5.30·13-s − 3.97·14-s + 0.0307·16-s + 2.88·17-s − 3.82·19-s + 4.35·20-s + 0.870·22-s + 7.78·23-s + 7.26·25-s + 4.61·26-s + 5.68·28-s − 0.514·29-s − 5.54·31-s + 5.67·32-s + 2.51·34-s + 16.0·35-s + 0.537·37-s − 3.32·38-s + 9.88·40-s + 3.66·41-s + ⋯ |
L(s) = 1 | + 0.615·2-s − 0.621·4-s − 1.56·5-s − 1.72·7-s − 0.997·8-s − 0.963·10-s + 0.301·11-s + 1.47·13-s − 1.06·14-s + 0.00769·16-s + 0.699·17-s − 0.877·19-s + 0.973·20-s + 0.185·22-s + 1.62·23-s + 1.45·25-s + 0.904·26-s + 1.07·28-s − 0.0954·29-s − 0.996·31-s + 1.00·32-s + 0.430·34-s + 2.70·35-s + 0.0883·37-s − 0.540·38-s + 1.56·40-s + 0.572·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 - 0.870T + 2T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 - 2.88T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 + 0.514T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 - 0.537T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 9.47T + 53T^{2} \) |
| 59 | \( 1 + 1.31T + 59T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 5.58T + 71T^{2} \) |
| 73 | \( 1 - 8.67T + 73T^{2} \) |
| 79 | \( 1 - 9.71T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−7.76731435882150810693678092422, −6.82060666873193095088450706209, −6.32783560194330437216003698216, −5.58745623825252950055659817902, −4.58777994002354743603081509018, −3.85425590319276147886659386972, −3.47520390210575986281771035046, −2.94969358317647020620764959414, −0.951717785297203355246956212909, 0,
0.951717785297203355246956212909, 2.94969358317647020620764959414, 3.47520390210575986281771035046, 3.85425590319276147886659386972, 4.58777994002354743603081509018, 5.58745623825252950055659817902, 6.32783560194330437216003698216, 6.82060666873193095088450706209, 7.76731435882150810693678092422