L(s) = 1 | − 2.57·2-s − 0.285·3-s + 4.61·4-s + 0.735·6-s − 7-s − 6.71·8-s − 2.91·9-s + 6.14·11-s − 1.31·12-s − 1.02·13-s + 2.57·14-s + 8.05·16-s − 0.880·17-s + 7.50·18-s − 3.91·19-s + 0.285·21-s − 15.7·22-s − 23-s + 1.92·24-s + 2.63·26-s + 1.69·27-s − 4.61·28-s + 6.92·29-s + 0.390·31-s − 7.26·32-s − 1.75·33-s + 2.26·34-s + ⋯ |
L(s) = 1 | − 1.81·2-s − 0.165·3-s + 2.30·4-s + 0.300·6-s − 0.377·7-s − 2.37·8-s − 0.972·9-s + 1.85·11-s − 0.380·12-s − 0.283·13-s + 0.687·14-s + 2.01·16-s − 0.213·17-s + 1.76·18-s − 0.898·19-s + 0.0623·21-s − 3.36·22-s − 0.208·23-s + 0.392·24-s + 0.516·26-s + 0.325·27-s − 0.871·28-s + 1.28·29-s + 0.0701·31-s − 1.28·32-s − 0.305·33-s + 0.388·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5630650985\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5630650985\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.285T + 3T^{2} \) |
| 11 | \( 1 - 6.14T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 17 | \( 1 + 0.880T + 17T^{2} \) |
| 19 | \( 1 + 3.91T + 19T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 0.390T + 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 41 | \( 1 + 3.45T + 41T^{2} \) |
| 43 | \( 1 + 0.0871T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 5.74T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 3.40T + 79T^{2} \) |
| 83 | \( 1 + 1.88T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.579437974079589788297167059327, −8.027346896412222374639510684189, −6.98952467748228492210471257123, −6.44998322429625650808189585884, −6.06973739313945582040468672702, −4.68792876435091145598951479537, −3.54041572035355247212918622103, −2.59488525114697427173494385993, −1.63802226858852342744940902416, −0.57490366203989930843298314199,
0.57490366203989930843298314199, 1.63802226858852342744940902416, 2.59488525114697427173494385993, 3.54041572035355247212918622103, 4.68792876435091145598951479537, 6.06973739313945582040468672702, 6.44998322429625650808189585884, 6.98952467748228492210471257123, 8.027346896412222374639510684189, 8.579437974079589788297167059327