L(s) = 1 | − 18.4i·5-s − 22.4·7-s − 53.6i·11-s + 7.15i·13-s − 39.6·17-s − 125. i·19-s + 99.1·23-s − 214.·25-s − 205. i·29-s + 147.·31-s + 413. i·35-s − 125. i·37-s − 506.·41-s − 413. i·43-s + 313.·47-s + ⋯ |
L(s) = 1 | − 1.64i·5-s − 1.21·7-s − 1.47i·11-s + 0.152i·13-s − 0.566·17-s − 1.51i·19-s + 0.898·23-s − 1.71·25-s − 1.31i·29-s + 0.854·31-s + 1.99i·35-s − 0.558i·37-s − 1.92·41-s − 1.46i·43-s + 0.974·47-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
\(L(2)\) |
\(\approx\) |
\(0.9221722803\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9221722803\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 18.4iT - 125T^{2} \) |
| 7 | \( 1 + 22.4T + 343T^{2} \) |
| 11 | \( 1 + 53.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 7.15iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 39.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 99.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 205. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 125. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 506.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 44.3iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 324iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 324iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 464. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 602.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 15.8iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 381.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 659.T + 9.12e5T^{2} \) |
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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
−8.774627265804539389111457098873, −8.550581806530188358520398482714, −7.18848770777239147019452303139, −6.28414900835992693853561856900, −5.47603610827312328089621019691, −4.60988110548407017380824839463, −3.62203917414064137035769306755, −2.52323450730434796435351532634, −0.867526806968042982994519562585, −0.27665851115955156014433845468,
1.79359288792865149497084368777, 2.93960701748030304612892091760, 3.49659026108066190713414331150, 4.72487291536099812897088799311, 6.03312569440941834672793385632, 6.78011085632551462362499837303, 7.10834555675639944607627271561, 8.163867011574369173971549215912, 9.422591686738293446086477076740, 10.05903426935536252610657612389