| L(s) = 1 | − 5-s − 1.73i·7-s − 1.48·11-s − 2.14·13-s + 3.22·17-s + 5.06i·19-s + (4.42 + 1.85i)23-s + 25-s − 5.21i·29-s − 5.45·31-s + 1.73i·35-s − 2.87i·37-s − 10.9i·41-s + 4.50i·43-s − 4.76i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.657i·7-s − 0.447·11-s − 0.595·13-s + 0.782·17-s + 1.16i·19-s + (0.922 + 0.387i)23-s + 0.200·25-s − 0.968i·29-s − 0.979·31-s + 0.294i·35-s − 0.473i·37-s − 1.71i·41-s + 0.686i·43-s − 0.694i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5840096201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5840096201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-4.42 - 1.85i)T \) |
| good | 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 - 5.06iT - 19T^{2} \) |
| 29 | \( 1 + 5.21iT - 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 + 2.87iT - 37T^{2} \) |
| 41 | \( 1 + 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 4.50iT - 43T^{2} \) |
| 47 | \( 1 + 4.76iT - 47T^{2} \) |
| 53 | \( 1 - 0.239T + 53T^{2} \) |
| 59 | \( 1 + 5.20iT - 59T^{2} \) |
| 61 | \( 1 - 6.30iT - 61T^{2} \) |
| 67 | \( 1 - 2.61iT - 67T^{2} \) |
| 71 | \( 1 + 3.17iT - 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 16.1iT - 79T^{2} \) |
| 83 | \( 1 - 0.583T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 1.66iT - 97T^{2} \) |
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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
−7.83939473340072834363038520884, −7.55389939812625052211638063331, −6.84881168893169548167378932561, −5.75055691541145409153057935282, −5.22071754089714882493664043782, −4.16434845098027879158463075632, −3.61032038967478758840698866173, −2.61805151317840533623493205116, −1.43533957843689917329909636909, −0.17532529439872040794072555064,
1.25396423838287920531885403963, 2.62399216783171204113838085493, 3.08454157054611197400939508097, 4.26972400141708631933205958942, 5.08115110722927161373486079573, 5.56948237247732069850457081677, 6.68868587934603153876134068774, 7.23993757980101850798166934787, 7.994193305187663121478012035338, 8.733037807588137465763074098356
