L(s) = 1 | − 1.31i·2-s + 9i·3-s + 30.2·4-s + (−35.6 − 43.0i)5-s + 11.8·6-s + 87.4i·7-s − 82.0i·8-s − 81·9-s + (−56.7 + 46.9i)10-s + 121·11-s + 272. i·12-s + 521. i·13-s + 115.·14-s + (387. − 320. i)15-s + 860.·16-s + 68.1i·17-s + ⋯ |
L(s) = 1 | − 0.232i·2-s + 0.577i·3-s + 0.945·4-s + (−0.637 − 0.770i)5-s + 0.134·6-s + 0.674i·7-s − 0.453i·8-s − 0.333·9-s + (−0.179 + 0.148i)10-s + 0.301·11-s + 0.546i·12-s + 0.855i·13-s + 0.157·14-s + (0.444 − 0.368i)15-s + 0.840·16-s + 0.0571i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.192543319\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.192543319\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9iT \) |
| 5 | \( 1 + (35.6 + 43.0i)T \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 + 1.31iT - 32T^{2} \) |
| 7 | \( 1 - 87.4iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 521. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 68.1iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.58e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 191. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 83.6T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.44e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 6.15e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.40e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.47e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.09e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.60e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.25e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.35e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 9.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.12e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.69e4iT - 8.58e9T^{2} \) |
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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
−11.77946589706944736433805433612, −11.38494247519712236322985062293, −10.00835770127684984731382630626, −9.060089461090847224560303118438, −8.009355869426122181597788600805, −6.72981538862580268093762608281, −5.44529258131571499011586771044, −4.17837098935044067764280194371, −2.86302567658525896242458274010, −1.25792157483903074419989700145,
0.802650503671775342022254272625, 2.52006836904914782789240651749, 3.67341697183289679044391147707, 5.61454416600208340396575571123, 6.83994847408242448081036458481, 7.38265970519279240186742975823, 8.281384428645557579022041489457, 10.09408981626091630481665034839, 10.96595571856152062826840855934, 11.72732120065056583200949159619