| L(s) = 1 | − 45.2·2-s − 512. i·3-s + 2.04e3·4-s + 1.46e4i·5-s + 2.32e4i·6-s − 9.26e4·8-s + 2.68e5·9-s − 6.63e5i·10-s − 2.68e6·11-s − 1.05e6i·12-s + 1.34e6i·13-s + 7.51e6·15-s + 4.19e6·16-s + 7.16e6i·17-s − 1.21e7·18-s − 7.31e7i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.703i·3-s + 0.500·4-s + 0.937i·5-s + 0.497i·6-s − 0.353·8-s + 0.505·9-s − 0.663i·10-s − 1.51·11-s − 0.351i·12-s + 0.279i·13-s + 0.659·15-s + 0.250·16-s + 0.296i·17-s − 0.357·18-s − 1.55i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.177394303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177394303\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 45.2T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 512. iT - 5.31e5T^{2} \) |
| 5 | \( 1 - 1.46e4iT - 2.44e8T^{2} \) |
| 11 | \( 1 + 2.68e6T + 3.13e12T^{2} \) |
| 13 | \( 1 - 1.34e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 - 7.16e6iT - 5.82e14T^{2} \) |
| 19 | \( 1 + 7.31e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 - 2.28e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + 5.69e8T + 3.53e17T^{2} \) |
| 31 | \( 1 - 3.08e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 - 1.70e9T + 6.58e18T^{2} \) |
| 41 | \( 1 - 5.37e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 9.19e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + 1.09e10iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 1.58e10T + 4.91e20T^{2} \) |
| 59 | \( 1 - 2.02e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 + 2.75e10iT - 2.65e21T^{2} \) |
| 67 | \( 1 - 6.70e10T + 8.18e21T^{2} \) |
| 71 | \( 1 - 7.34e10T + 1.64e22T^{2} \) |
| 73 | \( 1 - 6.49e10iT - 2.29e22T^{2} \) |
| 79 | \( 1 - 6.32e9T + 5.90e22T^{2} \) |
| 83 | \( 1 - 5.07e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 6.34e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 3.89e11iT - 6.93e23T^{2} \) |
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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
−11.32123702392721566844275805191, −10.66851738263763602431097940194, −9.575306438490227837851465319532, −8.215412849459077183889451149720, −7.19420087050129172316277059034, −6.65072865392128812417831740311, −5.02023442612863008751787580615, −3.05692099655143490953494912633, −2.16995207212414699126123671700, −0.805951750569038591324536837043,
0.44703373967940638767559885602, 1.68311350623491716100694082504, 3.23177006244575674944829227663, 4.70387610697292344586465737332, 5.59034436899190640965548223648, 7.35068461158992485552903553624, 8.289091583279842588299378935083, 9.363292752347499491192048522333, 10.19956205398452172366287492867, 11.05241772345534321748936865492
