| L(s) = 1 | + 27i·3-s + 467. i·5-s + 31.2·7-s − 729·9-s + 5.27e3i·11-s − 806. i·13-s − 1.26e4·15-s + 18.4·17-s − 1.36e4i·19-s + 844. i·21-s − 5.86e4·23-s − 1.40e5·25-s − 1.96e4i·27-s + 1.19e5i·29-s + 1.01e5·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.67i·5-s + 0.0344·7-s − 0.333·9-s + 1.19i·11-s − 0.101i·13-s − 0.965·15-s + 0.000908·17-s − 0.456i·19-s + 0.0198i·21-s − 1.00·23-s − 1.79·25-s − 0.192i·27-s + 0.907i·29-s + 0.614·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.4934733769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4934733769\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
| good | 5 | \( 1 - 467. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 31.2T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.27e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 806. iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 18.4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.36e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 5.86e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.19e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.01e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.59e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 8.09e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.53e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.83e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.07e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.43e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 8.84e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 2.48e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 4.68e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.38e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.29e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.60e6T + 8.07e13T^{2} \) |
| 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
−10.60551891661844952903581986548, −10.15623836008613351082656680482, −9.257658961508359303902251102222, −7.913331456347596084774259383283, −7.03687998920853890776783280178, −6.28537345663901660689383095334, −5.00279281369225306644668641171, −3.86920763230397216077056297617, −2.90983545614958247974336451689, −1.92799944693497375714297369255,
0.11051996825247442580790560839, 0.968217088121825989258314320937, 1.92587144695946162425599078921, 3.48932946923284946527851169800, 4.67218451000301770332380477041, 5.61790857695172023294179369773, 6.45124151531499947200874244997, 8.039544022315600179761649105768, 8.314899642993863078827772505941, 9.273003572386950536142318597403
