| L(s) = 1 | + (−2.82 + i)3-s + 6i·7-s + (7.00 − 5.65i)9-s + 5.65i·11-s − 10i·13-s − 22.6·17-s + 2·19-s + (−6 − 16.9i)21-s − 11.3·23-s + (−14.1 + 23.0i)27-s + 16.9i·29-s + 22·31-s + (−5.65 − 16.0i)33-s − 6i·37-s + (10 + 28.2i)39-s + ⋯ |
| L(s) = 1 | + (−0.942 + 0.333i)3-s + 0.857i·7-s + (0.777 − 0.628i)9-s + 0.514i·11-s − 0.769i·13-s − 1.33·17-s + 0.105·19-s + (−0.285 − 0.808i)21-s − 0.491·23-s + (−0.523 + 0.851i)27-s + 0.585i·29-s + 0.709·31-s + (−0.171 − 0.484i)33-s − 0.162i·37-s + (0.256 + 0.725i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4277449978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4277449978\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 - i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6iT - 49T^{2} \) |
| 11 | \( 1 - 5.65iT - 121T^{2} \) |
| 13 | \( 1 + 10iT - 169T^{2} \) |
| 17 | \( 1 + 22.6T + 289T^{2} \) |
| 19 | \( 1 - 2T + 361T^{2} \) |
| 23 | \( 1 + 11.3T + 529T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 - 22T + 961T^{2} \) |
| 37 | \( 1 + 6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 62.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86T + 3.72e3T^{2} \) |
| 67 | \( 1 + 2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 10T + 6.24e3T^{2} \) |
| 83 | \( 1 + 73.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94iT - 9.40e3T^{2} \) |
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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
−9.440850429807225734020280650415, −8.653411601402929081605180055196, −7.63265764443121731463548241099, −6.60913477039855778845591896451, −5.97473898376492286837498999175, −5.05577471395868185239380443949, −4.37888765129201213241982728690, −3.07566771995272463763786919740, −1.78914243077463375638658594177, −0.16456326129003196006968900426,
1.05465217151101387203679092698, 2.30926691989195602318665824048, 3.94759266142029296224158121254, 4.57185036419640137831163729311, 5.66343047953235140065319952628, 6.56768566136938710418945860978, 7.05442505115543255188234399041, 8.031546360634652157517920630330, 8.948475086866305329053735660512, 10.03177204033064718340350286208
