L(s) = 1 | + 1.73i·2-s − 1.73i·3-s − 0.999·4-s + 5-s + 2.99·6-s + (2 + 1.73i)7-s + 1.73i·8-s − 2.99·9-s + 1.73i·10-s − 3.46i·11-s + 1.73i·12-s + (−2.99 + 3.46i)14-s − 1.73i·15-s − 5·16-s − 6·17-s − 5.19i·18-s + ⋯ |
L(s) = 1 | + 1.22i·2-s − 0.999i·3-s − 0.499·4-s + 0.447·5-s + 1.22·6-s + (0.755 + 0.654i)7-s + 0.612i·8-s − 0.999·9-s + 0.547i·10-s − 1.04i·11-s + 0.499i·12-s + (−0.801 + 0.925i)14-s − 0.447i·15-s − 1.25·16-s − 1.45·17-s − 1.22i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02762 + 0.469472i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02762 + 0.469472i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−13.85073546011936671499247282926, −13.39455584922365110905441252970, −11.77580529432819705656530642586, −11.09076564889229001503688653183, −8.888675227619284864770247112347, −8.331081264168999936219915918912, −7.08173325171240219905876961637, −6.12624780347219580393116659172, −5.19569873787941261107632649192, −2.30989461230611783328334420826,
2.12350520376179601463862809320, 3.89188085758447586938992309253, 4.90641769224314713543468516814, 6.84319041448808681405548349722, 8.635844725493169134753519723351, 9.811044810689386859596554820332, 10.51196380823766213115919406936, 11.24803574257886029889692469881, 12.33787647325264601195125203552, 13.51046139945140927605779713303