L(s) = 1 | + (−1.41 − i)3-s + 2.82·5-s + 2.82i·7-s + (1.00 + 2.82i)9-s + 2i·11-s − 4i·13-s + (−4.00 − 2.82i)15-s + 5.65i·17-s + 2.82·19-s + (2.82 − 4.00i)21-s + 8·23-s + 3.00·25-s + (1.41 − 5.00i)27-s + 2.82·29-s − 8.48i·31-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.577i)3-s + 1.26·5-s + 1.06i·7-s + (0.333 + 0.942i)9-s + 0.603i·11-s − 1.10i·13-s + (−1.03 − 0.730i)15-s + 1.37i·17-s + 0.648·19-s + (0.617 − 0.872i)21-s + 1.66·23-s + 0.600·25-s + (0.272 − 0.962i)27-s + 0.525·29-s − 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30749 + 0.111352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30749 + 0.111352i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−11.39710785491590678053214760678, −10.42540176342404056017053440804, −9.714848876048339711521911435546, −8.613239515071452055373031359784, −7.51722692481107305037871800976, −6.27151545042709666373914242059, −5.73925680093140389530464301458, −4.90262128472470923969857010055, −2.73251475822297227269246547105, −1.54725662438713011073558818112,
1.15882525896736858807680651954, 3.17345858220063849604074900420, 4.62376623683424415542603396228, 5.38132132023293670107484485591, 6.56216569056071767764736280298, 7.15802566357469675757158148371, 9.014640719020964914143683714319, 9.547300273663319392966527971149, 10.46384748469546298881043016343, 11.12000131174485754522757097781