L(s) = 1 | − 2-s + i·3-s + 4-s + (−2.17 − 0.539i)5-s − i·6-s − 0.630·7-s − 8-s − 9-s + (2.17 + 0.539i)10-s − 3.07i·11-s + i·12-s + (2.87 − 2.17i)13-s + 0.630·14-s + (0.539 − 2.17i)15-s + 16-s − 3.70i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.970 − 0.241i)5-s − 0.408i·6-s − 0.238·7-s − 0.353·8-s − 0.333·9-s + (0.686 + 0.170i)10-s − 0.928i·11-s + 0.288i·12-s + (0.798 − 0.601i)13-s + 0.168·14-s + (0.139 − 0.560i)15-s + 0.250·16-s − 0.899i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.535067 - 0.353812i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.535067 - 0.353812i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.17 + 0.539i)T \) |
| 13 | \( 1 + (-2.87 + 2.17i)T \) |
good | 7 | \( 1 + 0.630T + 7T^{2} \) |
| 11 | \( 1 + 3.07iT - 11T^{2} \) |
| 17 | \( 1 + 3.70iT - 17T^{2} \) |
| 19 | \( 1 + 0.290iT - 19T^{2} \) |
| 23 | \( 1 + 1.70iT - 23T^{2} \) |
| 29 | \( 1 - 0.447T + 29T^{2} \) |
| 31 | \( 1 + 6.68iT - 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 5.75iT - 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 + 14.0iT - 53T^{2} \) |
| 59 | \( 1 - 9.02iT - 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 - 1.65iT - 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 3.23T + 83T^{2} \) |
| 89 | \( 1 - 12.3iT - 89T^{2} \) |
| 97 | \( 1 + 8.20T + 97T^{2} \) |
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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
−11.19273169763895204180295123915, −10.23512825022561796458774710725, −9.214454406603735495467680192372, −8.440832421676011869792809173111, −7.72473893680566216771478497891, −6.47860038903327482563555876290, −5.33066338832679045462462377031, −3.95279575150939540275416391622, −2.95874401367370896424782214228, −0.56287584350859621638684160636,
1.54114778183158116704486577621, 3.16175894264660280978586602361, 4.43858772875803094844806083539, 6.16256376006583740012697109310, 6.94964091104125596297336542112, 7.80964979834511407520257027043, 8.564490984054016487467588894698, 9.580622700475346066511898112344, 10.68903485152492507474785695882, 11.39066421169809790118300598362