L(s) = 1 | − 2.60·2-s − 0.581·3-s + 4.77·4-s + 3.23·5-s + 1.51·6-s − 2.22·7-s − 7.22·8-s − 2.66·9-s − 8.40·10-s + 4.03·11-s − 2.77·12-s − 4.71·13-s + 5.79·14-s − 1.87·15-s + 9.24·16-s − 2.68·17-s + 6.92·18-s + 5.82·19-s + 15.4·20-s + 1.29·21-s − 10.4·22-s − 3.12·23-s + 4.20·24-s + 5.43·25-s + 12.2·26-s + 3.29·27-s − 10.6·28-s + ⋯ |
L(s) = 1 | − 1.84·2-s − 0.335·3-s + 2.38·4-s + 1.44·5-s + 0.618·6-s − 0.841·7-s − 2.55·8-s − 0.887·9-s − 2.65·10-s + 1.21·11-s − 0.801·12-s − 1.30·13-s + 1.54·14-s − 0.485·15-s + 2.31·16-s − 0.652·17-s + 1.63·18-s + 1.33·19-s + 3.44·20-s + 0.282·21-s − 2.23·22-s − 0.650·23-s + 0.857·24-s + 1.08·25-s + 2.40·26-s + 0.633·27-s − 2.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6370881511\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6370881511\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4019 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 0.581T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 - 4.13T + 37T^{2} \) |
| 41 | \( 1 - 2.52T + 41T^{2} \) |
| 43 | \( 1 + 8.22T + 43T^{2} \) |
| 47 | \( 1 - 3.83T + 47T^{2} \) |
| 53 | \( 1 + 5.42T + 53T^{2} \) |
| 59 | \( 1 - 3.04T + 59T^{2} \) |
| 61 | \( 1 - 5.32T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 - 7.47T + 71T^{2} \) |
| 73 | \( 1 - 0.456T + 73T^{2} \) |
| 79 | \( 1 - 8.71T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.828792920975655400512831180747, −7.78512553304396377500236154651, −6.98612791071662479629519309069, −6.42015533814189998506356278858, −5.92006740273329889862524312363, −5.05023823162111616382391618388, −3.36145979469378182601401357125, −2.46997895595387086279613725691, −1.77867308477231643962060411847, −0.59794807919077894515599692099,
0.59794807919077894515599692099, 1.77867308477231643962060411847, 2.46997895595387086279613725691, 3.36145979469378182601401357125, 5.05023823162111616382391618388, 5.92006740273329889862524312363, 6.42015533814189998506356278858, 6.98612791071662479629519309069, 7.78512553304396377500236154651, 8.828792920975655400512831180747