L(s) = 1 | + 3i·3-s − 15.4·5-s + (6.29 − 17.4i)7-s − 9·9-s + 16.8·11-s − 9.10·13-s − 46.2i·15-s − 73.9i·17-s − 151. i·19-s + (52.2 + 18.8i)21-s + 0.264i·23-s + 112.·25-s − 27i·27-s + 279. i·29-s + 147.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.37·5-s + (0.339 − 0.940i)7-s − 0.333·9-s + 0.462·11-s − 0.194·13-s − 0.795i·15-s − 1.05i·17-s − 1.82i·19-s + (0.543 + 0.196i)21-s + 0.00239i·23-s + 0.897·25-s − 0.192i·27-s + 1.79i·29-s + 0.856·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0847i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1250705864\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1250705864\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-6.29 + 17.4i)T \) |
good | 5 | \( 1 + 15.4T + 125T^{2} \) |
| 11 | \( 1 - 16.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 9.10T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 151. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 0.264iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 279. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 418. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 164. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 266.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 276. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 212. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 174.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 317.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 106. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 755. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 194. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 997. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 988. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3iT - 9.12e5T^{2} \) |
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
−8.808584921869782619364651185777, −8.016670956806829914570750546842, −7.16990888081302375025480921230, −6.68957036948303402664142741654, −4.91843069834279108677089342461, −4.67587586483228674814549579316, −3.65578332366939664282444561732, −2.88485485956246468990407085110, −1.02023516904138885934217074165, −0.03491710065921986932522209886,
1.38084657097841086356769555995, 2.48434882350396127277752017946, 3.73291562250696377926483074782, 4.36004915353668245069422045797, 5.74248639746295871144917702006, 6.25241599694863412073166503747, 7.52057929885484311150647324466, 7.999606608386596498855451676661, 8.537604339367998583234226878471, 9.529370461428343790975554304274