L(s) = 1 | − 1.41·2-s + 2.00·4-s + (4.99 + 0.0147i)5-s + 5.91i·7-s − 2.82·8-s + (−7.07 − 0.0209i)10-s + 3.80i·11-s − 22.2i·13-s − 8.37i·14-s + 4.00·16-s − 1.20·17-s + 29.3·19-s + (9.99 + 0.0295i)20-s − 5.38i·22-s − 16.1·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s + (0.999 + 0.00295i)5-s + 0.845i·7-s − 0.353·8-s + (−0.707 − 0.00209i)10-s + 0.346i·11-s − 1.70i·13-s − 0.597i·14-s + 0.250·16-s − 0.0708·17-s + 1.54·19-s + (0.499 + 0.00147i)20-s − 0.244i·22-s − 0.701·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00295i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.683426961\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.683426961\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.99 - 0.0147i)T \) |
good | 7 | \( 1 - 5.91iT - 49T^{2} \) |
| 11 | \( 1 - 3.80iT - 121T^{2} \) |
| 13 | \( 1 + 22.2iT - 169T^{2} \) |
| 17 | \( 1 + 1.20T + 289T^{2} \) |
| 19 | \( 1 - 29.3T + 361T^{2} \) |
| 23 | \( 1 + 16.1T + 529T^{2} \) |
| 29 | \( 1 + 45.2iT - 841T^{2} \) |
| 31 | \( 1 - 43.7T + 961T^{2} \) |
| 37 | \( 1 - 48.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 2.72iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 19.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 24.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 10.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 19.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 18.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 83.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 121. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 105. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 94.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 94.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 132. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 119. iT - 9.40e3T^{2} \) |
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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
−9.864233700841012595639180759896, −9.462399863825466234011351739252, −8.294272872650435280844994931161, −7.77734136722072186336218047822, −6.46602929134701859473659357083, −5.77283764738460332016242037451, −4.97910704833368313895445260400, −3.11937880149611837768507817219, −2.31581197621584133844560065183, −0.940374532615153761942675661547,
1.02448329330649477667799943123, 2.06125778886661761199035611486, 3.42695441041899184758757281797, 4.70404356677814480297188391873, 5.84072060070861677369714573205, 6.77470280558553006312280618880, 7.34935972327070404718574800009, 8.560693147810053422803586814667, 9.321262840357779635152542751661, 9.914058150661941690968751163745