| L(s) = 1 | − 0.339·3-s − i·5-s − 3.88i·7-s − 2.88·9-s − 1.54i·11-s + (3.54 − 0.660i)13-s + 0.339i·15-s − 2.86i·19-s + 1.32i·21-s + 5.42·23-s − 25-s + 2·27-s − 5.20·29-s − 6.22i·31-s + 0.524i·33-s + ⋯ |
| L(s) = 1 | − 0.196·3-s − 0.447i·5-s − 1.46i·7-s − 0.961·9-s − 0.465i·11-s + (0.983 − 0.183i)13-s + 0.0877i·15-s − 0.657i·19-s + 0.288i·21-s + 1.13·23-s − 0.200·25-s + 0.384·27-s − 0.966·29-s − 1.11i·31-s + 0.0913i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.816246 - 0.678268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.816246 - 0.678268i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (-3.54 + 0.660i)T \) |
| good | 3 | \( 1 + 0.339T + 3T^{2} \) |
| 7 | \( 1 + 3.88iT - 7T^{2} \) |
| 11 | \( 1 + 1.54iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.86iT - 19T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 + 6.22iT - 31T^{2} \) |
| 37 | \( 1 - 8.56iT - 37T^{2} \) |
| 41 | \( 1 - 9.08iT - 41T^{2} \) |
| 43 | \( 1 - 0.980T + 43T^{2} \) |
| 47 | \( 1 - 6.52iT - 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 4.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 6.97iT - 67T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 - 3.43iT - 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8.56iT - 83T^{2} \) |
| 89 | \( 1 - 17.1iT - 89T^{2} \) |
| 97 | \( 1 + 13.7iT - 97T^{2} \) |
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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
−11.30438393746008789850740416108, −11.19406063249565506796310157420, −9.936873445749874374708830575235, −8.806296291302671338372418577886, −7.937729835859477063608209308651, −6.76723577313674734978808031095, −5.67626480644896880904916524832, −4.42694601665224417265258606945, −3.20869582271191960041843794520, −0.894464998644653211329237356339,
2.22453243916644613698578873050, 3.52194865543535629673090087107, 5.33706374066850246774696629849, 5.97334070312009896843874015217, 7.18723453642744003785268569533, 8.626837606767059098802184003958, 9.046120503917930060473072680772, 10.46303123223492261551361521557, 11.34564934986328715922268367728, 12.06479032731554021374154377572
