| L(s) = 1 | + 1.08i·3-s + 4.19i·7-s + 1.81·9-s − 6.43·11-s − 2.24i·13-s − 7.84i·17-s + 19-s − 4.57·21-s + 0.859i·23-s + 5.24i·27-s − 8.38·29-s + 1.24·31-s − 7.00i·33-s − 6.79i·37-s + 2.44·39-s + ⋯ |
| L(s) = 1 | + 0.629i·3-s + 1.58i·7-s + 0.603·9-s − 1.93·11-s − 0.622i·13-s − 1.90i·17-s + 0.229·19-s − 0.998·21-s + 0.179i·23-s + 1.00i·27-s − 1.55·29-s + 0.223·31-s − 1.22i·33-s − 1.11i·37-s + 0.392·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8435989445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8435989445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.08iT - 3T^{2} \) |
| 7 | \( 1 - 4.19iT - 7T^{2} \) |
| 11 | \( 1 + 6.43T + 11T^{2} \) |
| 13 | \( 1 + 2.24iT - 13T^{2} \) |
| 17 | \( 1 + 7.84iT - 17T^{2} \) |
| 23 | \( 1 - 0.859iT - 23T^{2} \) |
| 29 | \( 1 + 8.38T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 6.79iT - 37T^{2} \) |
| 41 | \( 1 - 5.92T + 41T^{2} \) |
| 43 | \( 1 + 6.81iT - 43T^{2} \) |
| 47 | \( 1 + 6.00iT - 47T^{2} \) |
| 53 | \( 1 + 13.7iT - 53T^{2} \) |
| 59 | \( 1 + 6.88T + 59T^{2} \) |
| 61 | \( 1 + 1.31T + 61T^{2} \) |
| 67 | \( 1 - 4.73iT - 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 9.86iT - 73T^{2} \) |
| 79 | \( 1 + 6.47T + 79T^{2} \) |
| 83 | \( 1 + 5.42iT - 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 - 6.76iT - 97T^{2} \) |
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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
−8.440773988081311533852614542088, −7.61120094381915172573762693235, −7.11616633629678418549954499275, −5.72345979791264051846318417021, −5.31948530712804123793568269655, −4.90192732373527555253126150981, −3.61474622168205322688214387459, −2.72423245908821145299813662791, −2.15782546972017060082210229081, −0.25451196312797026086849428210,
1.13019732281063635205870847342, 1.96328829706844614518156834605, 3.14693550359409801359083086715, 4.16752954955245966370696289108, 4.63150280078013734244886882671, 5.82972962624004847675359989668, 6.48800140691425511602149395286, 7.40769020051182964817194203170, 7.71328218689209946290811324007, 8.256046717671039017176452852842
