L(s) = 1 | − 0.715·2-s + 2.55·3-s − 1.48·4-s + 3.99·5-s − 1.82·6-s + 1.09·7-s + 2.49·8-s + 3.50·9-s − 2.85·10-s + 5.28·11-s − 3.79·12-s − 1.15·13-s − 0.784·14-s + 10.1·15-s + 1.18·16-s + 4.03·17-s − 2.51·18-s − 3.45·19-s − 5.93·20-s + 2.79·21-s − 3.78·22-s − 1.20·23-s + 6.36·24-s + 10.9·25-s + 0.824·26-s + 1.30·27-s − 1.63·28-s + ⋯ |
L(s) = 1 | − 0.506·2-s + 1.47·3-s − 0.743·4-s + 1.78·5-s − 0.745·6-s + 0.414·7-s + 0.882·8-s + 1.16·9-s − 0.903·10-s + 1.59·11-s − 1.09·12-s − 0.319·13-s − 0.209·14-s + 2.63·15-s + 0.297·16-s + 0.977·17-s − 0.592·18-s − 0.792·19-s − 1.32·20-s + 0.610·21-s − 0.806·22-s − 0.250·23-s + 1.30·24-s + 2.18·25-s + 0.161·26-s + 0.250·27-s − 0.308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.036818506\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.036818506\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2011 | \( 1 - T \) |
good | 2 | \( 1 + 0.715T + 2T^{2} \) |
| 3 | \( 1 - 2.55T + 3T^{2} \) |
| 5 | \( 1 - 3.99T + 5T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 - 4.03T + 17T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 + 6.62T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 + 9.60T + 47T^{2} \) |
| 53 | \( 1 - 2.26T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 + 2.06T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 - 3.86T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.161652813897252391049553640511, −8.631001304723881583866477231360, −7.941675521989985518020894314179, −6.95475448314924615161149503498, −6.02975587267437746478795053457, −5.07931285901519195772978436226, −4.10317890537437741288005867693, −3.17711772204561732650409095499, −1.89118670877148241594827744763, −1.46949269582407068108919217133,
1.46949269582407068108919217133, 1.89118670877148241594827744763, 3.17711772204561732650409095499, 4.10317890537437741288005867693, 5.07931285901519195772978436226, 6.02975587267437746478795053457, 6.95475448314924615161149503498, 7.941675521989985518020894314179, 8.631001304723881583866477231360, 9.161652813897252391049553640511