| L(s) = 1 | − i·2-s − 4-s + (2 + i)5-s − 2i·7-s + i·8-s + (1 − 2i)10-s − 2i·13-s − 2·14-s + 16-s + 6i·17-s + 6·19-s + (−2 − i)20-s + 4i·23-s + (3 + 4i)25-s − 2·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.894 + 0.447i)5-s − 0.755i·7-s + 0.353i·8-s + (0.316 − 0.632i)10-s − 0.554i·13-s − 0.534·14-s + 0.250·16-s + 1.45i·17-s + 1.37·19-s + (−0.447 − 0.223i)20-s + 0.834i·23-s + (0.600 + 0.800i)25-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.163539684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.163539684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 37 | \( 1 + iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.790657507695899428866478655475, −7.70106682266649789282781523532, −7.28955270036009381374914693300, −6.05841101543560902377254284994, −5.66411258571111249157217144678, −4.61898353245071880271270556702, −3.64405020535402858510681226182, −3.02000071470931212689766403768, −1.88364351669298361422952039795, −1.03682854773955450066706998918,
0.811057429414208159351577834397, 2.13280588464309931604440924500, 3.01007407333084677832434652189, 4.30914471504985980721897619817, 5.17003398075711093351289336321, 5.55091448086706236854686270203, 6.41104308799569143665203936482, 7.12730690892217117842173077894, 7.892535654995500238715883309690, 8.874011118790135662192664879321
