L(s) = 1 | − 5.15i·2-s − 3i·3-s − 18.5·4-s + (−7.01 − 8.70i)5-s − 15.4·6-s + 17.2i·7-s + 54.3i·8-s − 9·9-s + (−44.8 + 36.1i)10-s + 11·11-s + 55.6i·12-s − 3.57i·13-s + 88.7·14-s + (−26.1 + 21.0i)15-s + 131.·16-s − 101. i·17-s + ⋯ |
L(s) = 1 | − 1.82i·2-s − 0.577i·3-s − 2.31·4-s + (−0.627 − 0.778i)5-s − 1.05·6-s + 0.929i·7-s + 2.40i·8-s − 0.333·9-s + (−1.41 + 1.14i)10-s + 0.301·11-s + 1.33i·12-s − 0.0762i·13-s + 1.69·14-s + (−0.449 + 0.362i)15-s + 2.06·16-s − 1.45i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.268501 + 0.0947773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.268501 + 0.0947773i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 + (7.01 + 8.70i)T \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + 5.15iT - 8T^{2} \) |
| 7 | \( 1 - 17.2iT - 343T^{2} \) |
| 13 | \( 1 + 3.57iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 101. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 87.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 123. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 26.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 174.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 329. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 21.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 363. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 334. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 649. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 755.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 548. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 629.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.20e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 469.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 508. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 213. iT - 9.12e5T^{2} \) |
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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
−11.72465594327676271924984823186, −10.86086237513217079054729387992, −9.268799295244983065876176828686, −8.952343704744601686504077118463, −7.61863115632475210714482734999, −5.55441167092302402305513395107, −4.35900991124494541556642744443, −2.96150501077077573160270006129, −1.62894295302352482478440229700, −0.13637983183883384860225680541,
3.78663951588609388672515330206, 4.53315172978266056615306629391, 6.15515448618203795292100128892, 6.86908284235655178265177950074, 7.956691278216434153043994882943, 8.731203607266917837729677146304, 10.15068783580373347103615431567, 10.90189419483508150378450824473, 12.58354332576453789298818948941, 13.76210758262568180714132887823