L(s) = 1 | + 15.5i·3-s + 32·4-s − 243·9-s + 498. i·12-s + 1.14e3i·13-s + 1.02e3·16-s + 161. i·19-s − 3.12e3·25-s − 3.78e3i·27-s + 1.03e4i·31-s − 7.77e3·36-s + 6.66e3·37-s − 1.77e4·39-s − 2.24e4·43-s + 1.59e4i·48-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + 4-s − 9-s + 0.999i·12-s + 1.87i·13-s + 16-s + 0.102i·19-s − 25-s − 1.00i·27-s + 1.93i·31-s − 36-s + 0.799·37-s − 1.87·39-s − 1.85·43-s + 0.999i·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.946943969\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.946943969\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 15.5iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 32T^{2} \) |
| 5 | \( 1 + 3.12e3T^{2} \) |
| 11 | \( 1 - 1.61e5T^{2} \) |
| 13 | \( 1 - 1.14e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e6T^{2} \) |
| 19 | \( 1 - 161. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.03e4iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.24e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.34e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.80e9T^{2} \) |
| 73 | \( 1 + 4.67e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.27e5iT - 8.58e9T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.05209667589065974776508992130, −11.49734065026930261320698469912, −10.52176348900673877468694182112, −9.568641317143513811998128737777, −8.487735949416690918645448220058, −7.05305830452699616447645318448, −6.02292028923262869382215871363, −4.61818483850676143900332679746, −3.33781404314844089602731556852, −1.86243862887864202761484032072,
0.61319383602466123003363729961, 2.10094004331203923642477091921, 3.24023912076418603559457399214, 5.53488486678782408201045875137, 6.36362963533110108365532922463, 7.61818635230342723819810588003, 8.128821895189613885053504448728, 9.871762248068828598134417346377, 11.00677069816009224198216718914, 11.80383264807515841549356492128