L(s) = 1 | − 1.73i·3-s − 5.19i·7-s − 2.99·9-s − 7·13-s + 5.19i·19-s − 9·21-s + 5.19i·27-s − 1.73i·31-s + 10·37-s + 12.1i·39-s + 1.73i·43-s − 20·49-s + 9·57-s − 61-s + 15.5i·63-s + ⋯ |
L(s) = 1 | − 0.999i·3-s − 1.96i·7-s − 0.999·9-s − 1.94·13-s + 1.19i·19-s − 1.96·21-s + 0.999i·27-s − 0.311i·31-s + 1.64·37-s + 1.94i·39-s + 0.264i·43-s − 2.85·49-s + 1.19·57-s − 0.128·61-s + 1.96i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6634787623\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6634787623\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 5.19iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 19T + 97T^{2} \) |
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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
−9.422550516650503380920603643725, −7.953997253661533954297839297987, −7.65386470864920335632622302971, −6.98435933269876262489796397732, −6.13952396670763948482909527941, −4.92828676989730344467296981644, −4.01857656322000945710657715467, −2.82239737836733782608150438975, −1.53537794213936110850949704367, −0.27172348138090443192381610246,
2.47332103456048729065430526665, 2.80030907518900402726418495515, 4.40213043134028745468022780774, 5.15716719821352489369661815286, 5.71484093423719012499150660632, 6.83269136474204915474153533449, 8.010600507619943124845136573336, 8.835027303624975240967525120916, 9.448549833484785043448176587654, 9.916155051522485848918594032273