L(s) = 1 | + 28.5·3-s + (−5.81 − 55.5i)5-s + 92.3i·7-s + 571.·9-s + 541. i·11-s + 782.·13-s + (−165. − 1.58e3i)15-s + 1.55e3i·17-s + 484. i·19-s + 2.63e3i·21-s − 1.09e3i·23-s + (−3.05e3 + 646. i)25-s + 9.37e3·27-s + 597. i·29-s − 8.86e3·31-s + ⋯ |
L(s) = 1 | + 1.83·3-s + (−0.104 − 0.994i)5-s + 0.712i·7-s + 2.35·9-s + 1.34i·11-s + 1.28·13-s + (−0.190 − 1.82i)15-s + 1.30i·17-s + 0.307i·19-s + 1.30i·21-s − 0.432i·23-s + (−0.978 + 0.206i)25-s + 2.47·27-s + 0.131i·29-s − 1.65·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(4.449182718\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.449182718\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (5.81 + 55.5i)T \) |
good | 3 | \( 1 - 28.5T + 243T^{2} \) |
| 7 | \( 1 - 92.3iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 541. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 782.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.55e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 484. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.09e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 597. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 8.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.44e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.45e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.06e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.77e4iT - 8.58e9T^{2} \) |
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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
−10.58415402367294274897073269328, −9.467774888278446811832065245939, −8.929563752112982245248557880794, −8.232078782099206244138350361673, −7.44307318777659801697589235617, −5.93633499897389256597849230513, −4.41623254759363890735470962921, −3.69340622144658444952959998486, −2.21820083244608168635939874310, −1.45644737573918076809809142163,
1.01307981331554902431355382655, 2.57837203431196001123498913481, 3.35439699359214894423777558755, 4.08648431596903245171667439305, 6.06723997894171510321315094874, 7.28578326276479880099462645516, 7.81726801503950987413970942185, 8.891242149272449688238729546903, 9.559298521187718021265602278915, 10.76773057347829149602018571040