L(s) = 1 | − 1.80·2-s − 3-s + 1.24·4-s + 2.17·5-s + 1.80·6-s + 1.36·8-s + 9-s − 3.92·10-s − 0.798·11-s − 1.24·12-s − 1.19·13-s − 2.17·15-s − 4.94·16-s + 5.17·17-s − 1.80·18-s + 4.47·19-s + 2.70·20-s + 1.43·22-s + 6.71·23-s − 1.36·24-s − 0.257·25-s + 2.14·26-s − 27-s + 4.22·29-s + 3.92·30-s − 0.168·31-s + 6.17·32-s + ⋯ |
L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.621·4-s + 0.973·5-s + 0.735·6-s + 0.482·8-s + 0.333·9-s − 1.24·10-s − 0.240·11-s − 0.358·12-s − 0.330·13-s − 0.562·15-s − 1.23·16-s + 1.25·17-s − 0.424·18-s + 1.02·19-s + 0.605·20-s + 0.306·22-s + 1.40·23-s − 0.278·24-s − 0.0514·25-s + 0.421·26-s − 0.192·27-s + 0.785·29-s + 0.716·30-s − 0.0302·31-s + 1.09·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.054287441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.054287441\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 11 | \( 1 + 0.798T + 11T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 6.71T + 23T^{2} \) |
| 29 | \( 1 - 4.22T + 29T^{2} \) |
| 31 | \( 1 + 0.168T + 31T^{2} \) |
| 37 | \( 1 + 5.83T + 37T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 + 1.21T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 2.41T + 67T^{2} \) |
| 71 | \( 1 - 0.255T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 9.74T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−8.072796545632703177093429268621, −7.44016372724892161742390983067, −6.88962692996109684604702491790, −5.99619642247312477706576565205, −5.27200698804599718933796321746, −4.77657168746729053936155049115, −3.46543505772411709431546375742, −2.45278700060217463860993500330, −1.44172105402873126518928852879, −0.75024264805499257648189181871,
0.75024264805499257648189181871, 1.44172105402873126518928852879, 2.45278700060217463860993500330, 3.46543505772411709431546375742, 4.77657168746729053936155049115, 5.27200698804599718933796321746, 5.99619642247312477706576565205, 6.88962692996109684604702491790, 7.44016372724892161742390983067, 8.072796545632703177093429268621