L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 4i·7-s + i·8-s + 2·9-s + 3·11-s − i·12-s − 6i·13-s + 4·14-s + 16-s + 3i·17-s − 2i·18-s + 3·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.51i·7-s + 0.353i·8-s + 0.666·9-s + 0.904·11-s − 0.288i·12-s − 1.66i·13-s + 1.06·14-s + 0.250·16-s + 0.727i·17-s − 0.471i·18-s + 0.688·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.843361976\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.843361976\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 9iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 7iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.513586498980729825819219297522, −8.638618522769519574778284329310, −8.110695008298353972780594541605, −6.88033817248109741873666490695, −5.75144613715731921575387212724, −5.27743838130420467019957704975, −4.20103433894376308811809837828, −3.34758873621072759611640860067, −2.46041829672432624719476292181, −1.20681830647875515906255982504,
0.833852108929357477717526717899, 1.80810012447177684224483361883, 3.66737697165784384641241849548, 4.21009375396793741679631550736, 5.08043550053647302817168555454, 6.43159532328913016191916839869, 6.99013220587774793039682841010, 7.23523559506786352035792212560, 8.175195054702522799394078126707, 9.247047833047386106735151942613