| L(s) = 1 | − 3i·3-s − 14i·5-s − 24·7-s − 9·9-s − 28i·11-s − 74i·13-s − 42·15-s + 82·17-s − 92i·19-s + 72i·21-s + 8·23-s − 71·25-s + 27i·27-s − 138i·29-s − 80·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.25i·5-s − 1.29·7-s − 0.333·9-s − 0.767i·11-s − 1.57i·13-s − 0.722·15-s + 1.16·17-s − 1.11i·19-s + 0.748i·21-s + 0.0725·23-s − 0.568·25-s + 0.192i·27-s − 0.883i·29-s − 0.463·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.029883447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.029883447\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| good | 5 | \( 1 + 14iT - 125T^{2} \) |
| 7 | \( 1 + 24T + 343T^{2} \) |
| 11 | \( 1 + 28iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 74iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 82T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 80T + 2.97e4T^{2} \) |
| 37 | \( 1 + 30iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 282T + 6.89e4T^{2} \) |
| 43 | \( 1 - 4iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 240T + 1.03e5T^{2} \) |
| 53 | \( 1 - 130iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 596iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 218iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 436iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 856T + 3.57e5T^{2} \) |
| 73 | \( 1 - 998T + 3.89e5T^{2} \) |
| 79 | \( 1 - 32T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.50e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 246T + 7.04e5T^{2} \) |
| 97 | \( 1 - 866T + 9.12e5T^{2} \) |
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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
−9.357980000937952687519437417531, −8.468018738034087619685244699889, −7.83562745865748635318416457442, −6.71365559187858951590043620734, −5.74551550750571376064039165237, −5.15055055744073820492781780992, −3.61171907099952536160179163117, −2.76678432207595766815678875213, −0.984356794191576457958615555067, −0.33227264278853253182183223493,
1.92615398707818483023844590106, 3.25465306421631206924755681275, 3.73665523670741735984778100129, 5.09051760507480844054882651891, 6.36488920915821683270876777282, 6.76412950236055971932347983173, 7.73507435079042067658638917298, 9.062189721558620557559479501246, 9.843096620261683063123788637639, 10.19163377475394451099004904742
