L(s) = 1 | − 0.795·2-s − 3-s − 1.36·4-s − 1.31·5-s + 0.795·6-s + 2.67·8-s + 9-s + 1.04·10-s + 4.89·11-s + 1.36·12-s + 5.51·13-s + 1.31·15-s + 0.604·16-s + 3.91·17-s − 0.795·18-s + 4.67·19-s + 1.80·20-s − 3.89·22-s + 23-s − 2.67·24-s − 3.26·25-s − 4.38·26-s − 27-s + 6.49·29-s − 1.04·30-s + 6.46·31-s − 5.83·32-s + ⋯ |
L(s) = 1 | − 0.562·2-s − 0.577·3-s − 0.683·4-s − 0.589·5-s + 0.324·6-s + 0.946·8-s + 0.333·9-s + 0.331·10-s + 1.47·11-s + 0.394·12-s + 1.52·13-s + 0.340·15-s + 0.151·16-s + 0.950·17-s − 0.187·18-s + 1.07·19-s + 0.403·20-s − 0.830·22-s + 0.208·23-s − 0.546·24-s − 0.652·25-s − 0.860·26-s − 0.192·27-s + 1.20·29-s − 0.191·30-s + 1.16·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.131905213\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131905213\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 0.795T + 2T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 + 9.60T + 37T^{2} \) |
| 41 | \( 1 - 8.54T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 + 8.34T + 47T^{2} \) |
| 53 | \( 1 + 5.18T + 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 3.62T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 3.53T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 4.67T + 97T^{2} \) |
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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
−8.490474334922248700125989966751, −8.113917848534003908953331128371, −7.18576885722728769084880906042, −6.41159402765067570149876078990, −5.64090084692303572059591797550, −4.71382522365247990516829641650, −3.91797484737297443421344235192, −3.37174052532505500313705500003, −1.38158484291587584173960101139, −0.856026188860442099254797758065,
0.856026188860442099254797758065, 1.38158484291587584173960101139, 3.37174052532505500313705500003, 3.91797484737297443421344235192, 4.71382522365247990516829641650, 5.64090084692303572059591797550, 6.41159402765067570149876078990, 7.18576885722728769084880906042, 8.113917848534003908953331128371, 8.490474334922248700125989966751