| L(s) = 1 | − i·2-s − 4-s − 2.21·5-s + (2.36 − 1.18i)7-s + i·8-s + 2.21i·10-s + 4.88i·11-s + 0.107i·13-s + (−1.18 − 2.36i)14-s + 16-s + 0.672·17-s − 3.69i·19-s + 2.21·20-s + 4.88·22-s − 0.0322i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.990·5-s + (0.894 − 0.447i)7-s + 0.353i·8-s + 0.700i·10-s + 1.47i·11-s + 0.0297i·13-s + (−0.316 − 0.632i)14-s + 0.250·16-s + 0.163·17-s − 0.848i·19-s + 0.495·20-s + 1.04·22-s − 0.00671i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6400221951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6400221951\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.36 + 1.18i)T \) |
| 29 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.21T + 5T^{2} \) |
| 11 | \( 1 - 4.88iT - 11T^{2} \) |
| 13 | \( 1 - 0.107iT - 13T^{2} \) |
| 17 | \( 1 - 0.672T + 17T^{2} \) |
| 19 | \( 1 + 3.69iT - 19T^{2} \) |
| 23 | \( 1 + 0.0322iT - 23T^{2} \) |
| 31 | \( 1 - 2.41iT - 31T^{2} \) |
| 37 | \( 1 - 3.78T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 6.98T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 12.6iT - 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 4.53iT - 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 - 5.83iT - 71T^{2} \) |
| 73 | \( 1 + 0.289iT - 73T^{2} \) |
| 79 | \( 1 + 7.17T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 11.3iT - 97T^{2} \) |
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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
−8.726688176754531778332776034830, −7.889368287107619829521155165321, −7.44514611337122321296533677302, −6.71614320950834628193055757219, −5.37901670888803257019022025700, −4.54219345752730421459694777685, −4.26761515448261854519130881797, −3.22989444127614941346870810578, −2.16515895680613670156985185221, −1.21472140078882607913698477045,
0.20711025613768204578544082130, 1.56089189915028061615544801188, 3.05401761560102677754779945156, 3.80251590374468625282573296725, 4.61339314682079073814358098474, 5.51478354419803496687854078103, 6.00648915809918534715273856212, 7.00398710400982285753390634089, 7.82947491719360577370490965375, 8.269673169599329633068622843966
