| L(s) = 1 | − 4·4-s + 9i·7-s − 9·9-s + 16·16-s − 11i·17-s − 23i·23-s − 36i·28-s + 57·29-s − 53·31-s + 36·36-s − 51i·37-s − 33·41-s − 6i·43-s − 32·49-s − 101i·53-s + ⋯ |
| L(s) = 1 | − 4-s + 1.28i·7-s − 9-s + 16-s − 0.647i·17-s − i·23-s − 1.28i·28-s + 1.96·29-s − 1.70·31-s + 36-s − 1.37i·37-s − 0.804·41-s − 0.139i·43-s − 0.653·49-s − 1.90i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6278149582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6278149582\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + 23iT \) |
| good | 2 | \( 1 + 4T^{2} \) |
| 3 | \( 1 + 9T^{2} \) |
| 7 | \( 1 - 9iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 11iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 29 | \( 1 - 57T + 841T^{2} \) |
| 31 | \( 1 + 53T + 961T^{2} \) |
| 37 | \( 1 + 51iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33T + 1.68e3T^{2} \) |
| 43 | \( 1 + 6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 101iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 111iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 27T + 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 41iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 174iT - 9.40e3T^{2} \) |
| 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
−10.18639044708063414484515009731, −9.112746519857093525942293249269, −8.770644637692087626720310551335, −7.944348273142579988588109488481, −6.48787221812149731211365780830, −5.49290367521561014015759795903, −4.89131446991755701886211113823, −3.46152165171184711003592216770, −2.35155000208740894444951165322, −0.28379869387443741491947971854,
1.14843055427792614031553325437, 3.18167906418851621428924614159, 4.09429201901764397941036095313, 5.07282144531352294338782397822, 6.09654011047177270819405022452, 7.29017626822810836817548691053, 8.205352300985606674228640565642, 8.902425802114176384682052524904, 9.948904697411811935954663266421, 10.56958895106232392908395374872
