| L(s) = 1 | − i·2-s + 0.486i·3-s − 4-s + 2.55i·5-s + 0.486·6-s − 2.87·7-s + i·8-s + 2.76·9-s + 2.55·10-s − 11-s − 0.486i·12-s + 0.459·13-s + 2.87i·14-s − 1.24·15-s + 16-s − 6.17·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.281i·3-s − 0.5·4-s + 1.14i·5-s + 0.198·6-s − 1.08·7-s + 0.353i·8-s + 0.921·9-s + 0.807·10-s − 0.301·11-s − 0.140i·12-s + 0.127·13-s + 0.768i·14-s − 0.320·15-s + 0.250·16-s − 1.49·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3381770668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3381770668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| 53 | \( 1 + (3.74 - 6.24i)T \) |
| good | 3 | \( 1 - 0.486iT - 3T^{2} \) |
| 5 | \( 1 - 2.55iT - 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 13 | \( 1 - 0.459T + 13T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 + 0.299iT - 19T^{2} \) |
| 23 | \( 1 - 2.66iT - 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 + 5.49iT - 31T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 + 7.37iT - 41T^{2} \) |
| 43 | \( 1 + 7.00T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 7.52iT - 61T^{2} \) |
| 67 | \( 1 + 0.860iT - 67T^{2} \) |
| 71 | \( 1 - 0.429iT - 71T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 1.60iT - 79T^{2} \) |
| 83 | \( 1 + 9.15iT - 83T^{2} \) |
| 89 | \( 1 + 0.700T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−10.17166295785251475510103542775, −9.575333093071784970447715688536, −8.808278875388410541400874167192, −7.49404341145496859746349659468, −6.82081307178049358539479644573, −6.03693371472717306421154883158, −4.71972045363039185693557548189, −3.75374957742975272024131555957, −3.02714436477965303934216350277, −1.95308358187324017074585588979,
0.14117780043138217618750497277, 1.67194158716176200870484520125, 3.33585287380123421754813283850, 4.51805161558106871016927865173, 5.05397181945962888375056650394, 6.42739932821670560471471954907, 6.69204148789932872569100356034, 7.82827476086316517260438779665, 8.605629483440919116085063223924, 9.303765527206021343065986057827
