| L(s) = 1 | + 1.73·3-s + (−4.91 + 0.905i)5-s + (−1.91 + 6.73i)7-s + 2.99·9-s − 17.5·11-s − 4.83·13-s + (−8.51 + 1.56i)15-s + 18.0·17-s + 9.13i·19-s + (−3.31 + 11.6i)21-s + 3.72i·23-s + (23.3 − 8.90i)25-s + 5.19·27-s + 1.12·29-s + 57.0i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (−0.983 + 0.181i)5-s + (−0.273 + 0.961i)7-s + 0.333·9-s − 1.59·11-s − 0.371·13-s + (−0.567 + 0.104i)15-s + 1.06·17-s + 0.480i·19-s + (−0.157 + 0.555i)21-s + 0.161i·23-s + (0.934 − 0.356i)25-s + 0.192·27-s + 0.0388·29-s + 1.84i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0946 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0946 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5961915624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5961915624\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 + (4.91 - 0.905i)T \) |
| 7 | \( 1 + (1.91 - 6.73i)T \) |
| good | 11 | \( 1 + 17.5T + 121T^{2} \) |
| 13 | \( 1 + 4.83T + 169T^{2} \) |
| 17 | \( 1 - 18.0T + 289T^{2} \) |
| 19 | \( 1 - 9.13iT - 361T^{2} \) |
| 23 | \( 1 - 3.72iT - 529T^{2} \) |
| 29 | \( 1 - 1.12T + 841T^{2} \) |
| 31 | \( 1 - 57.0iT - 961T^{2} \) |
| 37 | \( 1 + 41.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 11.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 64.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 77.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 87.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.36iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 47.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 58.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 53.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 28.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 107.T + 9.40e3T^{2} \) |
| 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
−8.715232902506947802241793845110, −8.192251605940381998846553870220, −7.55638543991850662409067942688, −6.75577470423582880966855424777, −5.47278844181032956511657298764, −4.96464658321524018000278866340, −3.56030351462510160416065853061, −3.03793047536223325003908217150, −2.00400269932028868961032284001, −0.17037775170297148981261111819,
0.986950602010611349370418257089, 2.66073521555423946298139323938, 3.35084213635033272661126341534, 4.37566317557765722286666760578, 5.01018227461153377273153957355, 6.27301234333590720801247099119, 7.40953081680509881151799373613, 7.75497301825445315603326186841, 8.280904263412732318085636952316, 9.482331459963871334741916837325
