L(s) = 1 | + 1.69·3-s + 4.40i·7-s − 0.120·9-s − 0.720i·11-s + 4.59·13-s + 2.38i·17-s + 0.899i·19-s + 7.46i·21-s + 8.05i·23-s − 5.29·27-s − 1.78i·29-s − 9.07·31-s − 1.22i·33-s − 4.74·37-s + 7.79·39-s + ⋯ |
L(s) = 1 | + 0.979·3-s + 1.66i·7-s − 0.0403·9-s − 0.217i·11-s + 1.27·13-s + 0.578i·17-s + 0.206i·19-s + 1.62i·21-s + 1.67i·23-s − 1.01·27-s − 0.330i·29-s − 1.63·31-s − 0.212i·33-s − 0.780·37-s + 1.24·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.148398345\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.148398345\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.69T + 3T^{2} \) |
| 7 | \( 1 - 4.40iT - 7T^{2} \) |
| 11 | \( 1 + 0.720iT - 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 17 | \( 1 - 2.38iT - 17T^{2} \) |
| 19 | \( 1 - 0.899iT - 19T^{2} \) |
| 23 | \( 1 - 8.05iT - 23T^{2} \) |
| 29 | \( 1 + 1.78iT - 29T^{2} \) |
| 31 | \( 1 + 9.07T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4.99iT - 47T^{2} \) |
| 53 | \( 1 + 9.74T + 53T^{2} \) |
| 59 | \( 1 + 6.39iT - 59T^{2} \) |
| 61 | \( 1 - 0.635iT - 61T^{2} \) |
| 67 | \( 1 - 1.73T + 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 + 13.0iT - 73T^{2} \) |
| 79 | \( 1 - 7.19T + 79T^{2} \) |
| 83 | \( 1 + 0.219T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 - 8.41iT - 97T^{2} \) |
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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
−8.766910210058784673529713386662, −8.106811719684332737034114073362, −7.52884284130922128376699785576, −6.21831031446642844860736565154, −5.83417136886364490084892100762, −5.10907839620085837155125701921, −3.65263498301747521274655304023, −3.37961354872105226856549421058, −2.28390951731936740778495094967, −1.61487152866676114757520387578,
0.50791651635564959475238728122, 1.65884589923635969369981590569, 2.77109735933833708800738354002, 3.66228022943711735060469413120, 4.09113223512557071932871765888, 5.07713828601738376526324368983, 6.15125510772744547738034541603, 6.98938830175541934507054606091, 7.46153528315247202376143177887, 8.300312063525726526248557628276