| L(s) = 1 | − 3i·3-s − 16.7·5-s + (16.4 − 8.44i)7-s − 9·9-s + 8.98·11-s − 6.28·13-s + 50.1i·15-s − 106. i·17-s + 25.1i·19-s + (−25.3 − 49.4i)21-s + 19.5i·23-s + 154.·25-s + 27i·27-s − 132. i·29-s + 120.·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.49·5-s + (0.889 − 0.456i)7-s − 0.333·9-s + 0.246·11-s − 0.134·13-s + 0.862i·15-s − 1.52i·17-s + 0.304i·19-s + (−0.263 − 0.513i)21-s + 0.177i·23-s + 1.23·25-s + 0.192i·27-s − 0.850i·29-s + 0.699·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.306i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6604667560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6604667560\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (-16.4 + 8.44i)T \) |
| good | 5 | \( 1 + 16.7T + 125T^{2} \) |
| 11 | \( 1 - 8.98T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.28T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 25.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 19.5iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 132. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 72.1iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 410. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 64.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 401. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 584. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 736.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 643.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 186. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 64.5iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 50.5iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 17.9iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 378. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 593. iT - 9.12e5T^{2} \) |
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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
−8.547751288064680585497344804109, −7.76595752978030260511344111027, −7.44672837224336423061632071135, −6.59179087077341146165976981182, −5.28248541758039224874824026742, −4.45654926743533002599596570922, −3.65229419035014117384560978450, −2.48132795337083717556740555214, −1.08126305606312506975657750021, −0.18134913654580299313682482425,
1.33765751169616798800476741770, 2.79591393739756885599030716660, 3.88027018472373211711023278811, 4.43083295740997174042642372172, 5.31549659139874440938452505661, 6.40811316742997977971257810511, 7.45509215919561122756398752013, 8.286885657747364721475656582758, 8.568277050504938422585762801536, 9.636639431910829894322552791754
