L(s) = 1 | + 0.474i·2-s + i·3-s + 1.77·4-s − i·5-s − 0.474·6-s + (0.474 + 2.60i)7-s + 1.79i·8-s − 9-s + 0.474·10-s + (3.23 + 0.726i)11-s + 1.77i·12-s − 0.814·13-s + (−1.23 + 0.225i)14-s + 15-s + 2.69·16-s + 6.40·17-s + ⋯ |
L(s) = 1 | + 0.335i·2-s + 0.577i·3-s + 0.887·4-s − 0.447i·5-s − 0.193·6-s + (0.179 + 0.983i)7-s + 0.633i·8-s − 0.333·9-s + 0.150·10-s + (0.975 + 0.219i)11-s + 0.512i·12-s − 0.225·13-s + (−0.330 + 0.0602i)14-s + 0.258·15-s + 0.674·16-s + 1.55·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0403 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0403 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.166357519\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.166357519\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-0.474 - 2.60i)T \) |
| 11 | \( 1 + (-3.23 - 0.726i)T \) |
good | 2 | \( 1 - 0.474iT - 2T^{2} \) |
| 13 | \( 1 + 0.814T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 + 0.774T + 23T^{2} \) |
| 29 | \( 1 - 2.51iT - 29T^{2} \) |
| 31 | \( 1 + 5.42iT - 31T^{2} \) |
| 37 | \( 1 + 5.42T + 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 - 6.55iT - 43T^{2} \) |
| 47 | \( 1 - 4.57iT - 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 + 3.34iT - 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 + 3.04T + 71T^{2} \) |
| 73 | \( 1 + 9.23T + 73T^{2} \) |
| 79 | \( 1 + 7.97iT - 79T^{2} \) |
| 83 | \( 1 - 6.58T + 83T^{2} \) |
| 89 | \( 1 - 1.12iT - 89T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−9.902453412731272772805848301478, −9.153352714257387853816676378066, −8.352820276906183139173482560161, −7.61783476044990301564659382112, −6.47781655371294803007273764818, −5.81057440309018851296656617305, −5.03368860861312085141767889943, −3.88646980141058450540465285006, −2.72839039123051453017769031601, −1.60548767575954403437744871745,
1.00863072030114355832978182673, 2.04103866604268116015605706662, 3.28142841712379125061014864284, 4.01205462521742132049066568238, 5.55261540069626303493626919156, 6.51117981155083342869757422923, 7.04503022397032173151824268800, 7.72485167879666237953314448657, 8.662951478205005080140287496140, 9.945066628190083852783756706755