| L(s) = 1 | + 8i·5-s + 12·7-s + 12i·11-s − 20i·13-s − 62·17-s − 108i·19-s + 72·23-s + 61·25-s − 128i·29-s − 204·31-s + 96i·35-s − 228i·37-s + 22·41-s − 204i·43-s + 600·47-s + ⋯ |
| L(s) = 1 | + 0.715i·5-s + 0.647·7-s + 0.328i·11-s − 0.426i·13-s − 0.884·17-s − 1.30i·19-s + 0.652·23-s + 0.487·25-s − 0.819i·29-s − 1.18·31-s + 0.463i·35-s − 1.01i·37-s + 0.0838·41-s − 0.723i·43-s + 1.86·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.899880169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.899880169\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 8iT - 125T^{2} \) |
| 7 | \( 1 - 12T + 343T^{2} \) |
| 11 | \( 1 - 12iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 62T + 4.91e3T^{2} \) |
| 19 | \( 1 + 108iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 72T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 204T + 2.97e4T^{2} \) |
| 37 | \( 1 + 228iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 22T + 6.89e4T^{2} \) |
| 43 | \( 1 + 204iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 600T + 1.03e5T^{2} \) |
| 53 | \( 1 + 256iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 828iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 84iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 348iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 456T + 3.57e5T^{2} \) |
| 73 | \( 1 - 822T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.35e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 108iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 938T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.27e3T + 9.12e5T^{2} \) |
| 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
−9.177252523231468203718978657844, −8.641500388361952812324821211185, −7.38924875861704834676787760309, −7.07983997740716503887772709393, −5.95209827040311355225035888439, −4.99247061086554422537695258635, −4.12470892235785676545173370293, −2.89272935143573295511514797278, −2.04221940129302325136765615199, −0.51368626996470401308829487096,
1.03611510174584248445418526789, 1.97665524371833984611449933188, 3.38408538893782738089383259557, 4.45938509546728866066680399989, 5.14167766176179896237156440009, 6.10023390584808819485700224791, 7.09177982770107387331283098528, 8.011356051206735430009355390670, 8.756373790842805118871184503942, 9.289478690928624504344663680716
