L(s) = 1 | − 18.8i·2-s − 54.3·3-s − 228.·4-s + 103. i·5-s + 1.02e3i·6-s − 202.·7-s + 1.89e3i·8-s + 772.·9-s + 1.95e3·10-s + 2.66e3·11-s + 1.24e4·12-s + 336. i·13-s + 3.82e3i·14-s − 5.63e3i·15-s + 6.57e3·16-s + 5.88e3i·17-s + ⋯ |
L(s) = 1 | − 1.66i·2-s − 1.16·3-s − 1.78·4-s + 0.370i·5-s + 1.94i·6-s − 0.223·7-s + 1.30i·8-s + 0.353·9-s + 0.618·10-s + 0.602·11-s + 2.07·12-s + 0.0424i·13-s + 0.372i·14-s − 0.430i·15-s + 0.401·16-s + 0.290i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.605221 - 0.0827767i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.605221 - 0.0827767i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-2.96e5 - 8.27e4i)T \) |
good | 2 | \( 1 + 18.8iT - 128T^{2} \) |
| 3 | \( 1 + 54.3T + 2.18e3T^{2} \) |
| 5 | \( 1 - 103. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 202.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.66e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 336. iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 5.88e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.63e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 2.68e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.68e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.97e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 + 2.52e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.02e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 3.28e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.37e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.89e4iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 2.40e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.27e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.07e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.38e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.15e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 3.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.56e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.46e7iT - 8.07e13T^{2} \) |
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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
−14.44967041560743066172367086810, −13.01385229792324030716087989660, −12.04045332016627040415167847745, −11.17232289330389865498322651793, −10.39111863151358264637825957704, −9.031980147826384936607977139723, −6.59953785145121422426307552328, −4.85513326890721121807439162605, −3.15973766580433334089177874555, −1.19851139802019810175509621863,
0.37293719705110555720332956572, 4.55406718452798871546749004615, 5.79148536833348032373366880461, 6.65233558987585314855220099421, 8.143354664147934369323461371652, 9.555402577769884323380152124885, 11.33737301156204998990390050388, 12.66447554231510539389041132957, 14.03746754018559911528465024690, 15.19570012459004417228634719994