| L(s) = 1 | + 3.16·3-s + (1.58 − 4.74i)5-s + (−6.32 − 3i)7-s + 1.00·9-s − 14·11-s + 3.16·13-s + (5.00 − 15.0i)15-s − 6.32·17-s − 28.4i·19-s + (−20.0 − 9.48i)21-s − 12i·23-s + (−20 − 15.0i)25-s − 25.2·27-s + 14·29-s + 37.9i·31-s + ⋯ |
| L(s) = 1 | + 1.05·3-s + (0.316 − 0.948i)5-s + (−0.903 − 0.428i)7-s + 0.111·9-s − 1.27·11-s + 0.243·13-s + (0.333 − i)15-s − 0.372·17-s − 1.49i·19-s + (−0.952 − 0.451i)21-s − 0.521i·23-s + (−0.800 − 0.600i)25-s − 0.936·27-s + 0.482·29-s + 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.448331952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.448331952\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.58 + 4.74i)T \) |
| 7 | \( 1 + (6.32 + 3i)T \) |
| good | 3 | \( 1 - 3.16T + 9T^{2} \) |
| 11 | \( 1 + 14T + 121T^{2} \) |
| 13 | \( 1 - 3.16T + 169T^{2} \) |
| 17 | \( 1 + 6.32T + 289T^{2} \) |
| 19 | \( 1 + 28.4iT - 361T^{2} \) |
| 23 | \( 1 + 12iT - 529T^{2} \) |
| 29 | \( 1 - 14T + 841T^{2} \) |
| 31 | \( 1 - 37.9iT - 961T^{2} \) |
| 37 | \( 1 + 18iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 44.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 54iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 9.48iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 66.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 102iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 16T + 5.04e3T^{2} \) |
| 73 | \( 1 + 63.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 72.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 56.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 69.5T + 9.40e3T^{2} \) |
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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
−10.08747130445933471786476729439, −9.098720969285788687934909012081, −8.688968627075435949060928648871, −7.72902976866047530754817485991, −6.71090933668969609676393932667, −5.48470801740968221709066742283, −4.47892317734127450559411228053, −3.19792585246473343816506542262, −2.29815274893213798139575876993, −0.44252860729259933139155984956,
2.21176869459850239082712209521, 2.93375486564566336744836396032, 3.80216004714425403692442460676, 5.58849580934222708569091467168, 6.28955088966645892753930971211, 7.52084930229740584925511925029, 8.148975868672997126563532389782, 9.214486021857355593198915580035, 9.936324542584604403633937421408, 10.63093700806983754282988048361
