L(s) = 1 | + (1 + 1.41i)3-s + i·5-s + 4.82i·7-s + (−1.00 + 2.82i)9-s + 4.82·11-s − 1.17·13-s + (−1.41 + i)15-s + 0.828i·17-s − 6i·19-s + (−6.82 + 4.82i)21-s − 0.828·23-s − 25-s + (−5.00 + 1.41i)27-s − 6i·29-s − 2i·31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.816i)3-s + 0.447i·5-s + 1.82i·7-s + (−0.333 + 0.942i)9-s + 1.45·11-s − 0.324·13-s + (−0.365 + 0.258i)15-s + 0.200i·17-s − 1.37i·19-s + (−1.49 + 1.05i)21-s − 0.172·23-s − 0.200·25-s + (−0.962 + 0.272i)27-s − 1.11i·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.874124 + 1.68867i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.874124 + 1.68867i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 5 | \( 1 - iT \) |
good | 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 0.828iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 - 9.65iT - 41T^{2} \) |
| 43 | \( 1 - 1.17iT - 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 - 7.65iT - 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.48iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + 5.31T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 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
−10.03787099025669983471078530018, −9.370559096551686573636780119566, −8.858972766466422535601861433405, −8.112853174296022300909404223257, −6.83678509304477212891857999770, −5.96063623558706253229580618424, −5.03489596428867291242601805078, −4.01611478974788089121960696778, −2.89667550528127569788319818542, −2.13039623844281107634976251426,
0.867620499512216398263893718943, 1.77883526319165486956040410359, 3.64240215294587111789171267225, 3.95706193444316124565830539442, 5.42895977741174359168047527154, 6.87785912801087497102091757090, 6.94824347531736703492614614601, 8.048411941069532440827649044462, 8.770967888068998481182179720366, 9.713611661604259564824995298365