| L(s) = 1 | − 3-s − 3.60i·5-s + 4.49i·7-s + 9-s − 0.890i·11-s + (−2.60 − 2.49i)13-s + 3.60i·15-s + 2·17-s − 4.49i·19-s − 4.49i·21-s + 1.78·23-s − 7.98·25-s − 27-s + 0.219·29-s − 2.71i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.61i·5-s + 1.69i·7-s + 0.333·9-s − 0.268i·11-s + (−0.722 − 0.691i)13-s + 0.930i·15-s + 0.485·17-s − 1.03i·19-s − 0.980i·21-s + 0.371·23-s − 1.59·25-s − 0.192·27-s + 0.0408·29-s − 0.487i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7635537900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7635537900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + (2.60 + 2.49i)T \) |
| good | 5 | \( 1 + 3.60iT - 5T^{2} \) |
| 7 | \( 1 - 4.49iT - 7T^{2} \) |
| 11 | \( 1 + 0.890iT - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4.49iT - 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 - 0.219T + 29T^{2} \) |
| 31 | \( 1 + 2.71iT - 31T^{2} \) |
| 37 | \( 1 - 5.78iT - 37T^{2} \) |
| 41 | \( 1 + 10.8iT - 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 4.89iT - 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 8.09iT - 59T^{2} \) |
| 61 | \( 1 - 7.42T + 61T^{2} \) |
| 67 | \( 1 + 3.70iT - 67T^{2} \) |
| 71 | \( 1 - 5.87iT - 71T^{2} \) |
| 73 | \( 1 + 7.20iT - 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 14.3iT - 83T^{2} \) |
| 89 | \( 1 + 6.37iT - 89T^{2} \) |
| 97 | \( 1 + 15.2iT - 97T^{2} \) |
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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
−9.354151256700390494093625841666, −8.585815699697149350970832853490, −8.087218060842183336615837409458, −6.79593588526198519664702487314, −5.63765103915979621412444461246, −5.28591200177483758591412361802, −4.62298390077472667942315563942, −3.05992099045788156199742065127, −1.80844820495353530818596501844, −0.34963640523688639349913015103,
1.52415534834605723405463764158, 3.02858550914698454215562174970, 3.90294303401365030953616597655, 4.78335763028937990130234410775, 6.10446619025708903352272425997, 6.85238101965168061005490213946, 7.28396057694623619048148461560, 8.004763206662775685309945608944, 9.730906813354586651364417616176, 10.04361860787241988658289841460
