| L(s) = 1 | − i·2-s − 1.73·3-s − 4-s − i·5-s + 1.73i·6-s + 2·7-s + i·8-s + 2.99·9-s − 10-s − 3.46·11-s + 1.73·12-s − 2i·14-s + 1.73i·15-s + 16-s + 6.92·17-s − 2.99i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.00·3-s − 0.5·4-s − 0.447i·5-s + 0.707i·6-s + 0.755·7-s + 0.353i·8-s + 0.999·9-s − 0.316·10-s − 1.04·11-s + 0.500·12-s − 0.534i·14-s + 0.447i·15-s + 0.250·16-s + 1.68·17-s − 0.707i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.596792 - 0.869179i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.596792 - 0.869179i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + iT \) |
| 31 | \( 1 + (-2 + 5.19i)T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.16485121304328189420215584400, −9.229159396788466427933696977508, −7.943736658397663323892169643821, −7.58440142594767648017901814633, −5.99612074181969862511812857112, −5.27771491747592945946232803588, −4.65603259421119616742625761070, −3.43791087755009920403789701190, −1.90311921155653360308369447639, −0.67861203410769680658360009911,
1.22995218427396010918342341399, 3.08961569683736646268920397786, 4.48085670489492796905245735334, 5.30445845699822379114276333090, 5.87124436445662087023989862742, 6.94334253513056810876342563610, 7.71792313185852165486109446855, 8.264271537082183407924854445820, 9.834249739481280690320436171265, 10.15476420169740390528361202020
