| L(s) = 1 | + i·2-s − 4-s + i·5-s − 4.66i·7-s − i·8-s − 10-s + 4.89i·11-s + (−3.44 − 1.04i)13-s + 4.66·14-s + 16-s − 2.57·17-s + 6.76i·19-s − i·20-s − 4.89·22-s − 4.66·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447i·5-s − 1.76i·7-s − 0.353i·8-s − 0.316·10-s + 1.47i·11-s + (−0.956 − 0.291i)13-s + 1.24·14-s + 0.250·16-s − 0.623·17-s + 1.55i·19-s − 0.223i·20-s − 1.04·22-s − 0.973·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7459355093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7459355093\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (3.44 + 1.04i)T \) |
| good | 7 | \( 1 + 4.66iT - 7T^{2} \) |
| 11 | \( 1 - 4.89iT - 11T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 - 6.76iT - 19T^{2} \) |
| 23 | \( 1 + 4.66T + 23T^{2} \) |
| 29 | \( 1 - 6.76T + 29T^{2} \) |
| 31 | \( 1 - 2.09iT - 31T^{2} \) |
| 37 | \( 1 - 11.4iT - 37T^{2} \) |
| 41 | \( 1 - 10.8iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 9.79iT - 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 + 4.89iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 7.23iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 14.0iT - 73T^{2} \) |
| 79 | \( 1 + 5.79T + 79T^{2} \) |
| 83 | \( 1 + 9.79iT - 83T^{2} \) |
| 89 | \( 1 + 1.10iT - 89T^{2} \) |
| 97 | \( 1 - 9.80iT - 97T^{2} \) |
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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
−10.09678333004602110103621227185, −9.613904976701241590581795279449, −7.945580101838017077954314649259, −7.79943080903974405777162536320, −6.79935749903229508437889646469, −6.34544276636594407825714803077, −4.74012663648246216659686285947, −4.41308172340377291831094486086, −3.20883463928280451469154320965, −1.57247151604754220363885382819,
0.31492450708377922791651424534, 2.18886053978888912844667466711, 2.75753221647245771417506397499, 4.09134414408626696183448565811, 5.21080027865901388128111067279, 5.71138562181517760295129049248, 6.82177784645334299615038073036, 8.187519270488524025384271132811, 8.893107950827281635948143696539, 9.131402544614132337642404183231
