| L(s) = 1 | + 0.993·2-s + 1.33·3-s − 1.01·4-s − 3.92·5-s + 1.32·6-s + 1.61·7-s − 2.99·8-s − 1.21·9-s − 3.90·10-s − 11-s − 1.35·12-s − 1.90·13-s + 1.60·14-s − 5.24·15-s − 0.949·16-s − 1.20·18-s + 2.61·19-s + 3.97·20-s + 2.16·21-s − 0.993·22-s − 3.64·23-s − 4.00·24-s + 10.4·25-s − 1.89·26-s − 5.63·27-s − 1.63·28-s − 0.521·29-s + ⋯ |
| L(s) = 1 | + 0.702·2-s + 0.771·3-s − 0.506·4-s − 1.75·5-s + 0.542·6-s + 0.612·7-s − 1.05·8-s − 0.404·9-s − 1.23·10-s − 0.301·11-s − 0.390·12-s − 0.528·13-s + 0.430·14-s − 1.35·15-s − 0.237·16-s − 0.284·18-s + 0.600·19-s + 0.889·20-s + 0.472·21-s − 0.211·22-s − 0.759·23-s − 0.816·24-s + 2.08·25-s − 0.371·26-s − 1.08·27-s − 0.309·28-s − 0.0967·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.558305354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.558305354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.993T + 2T^{2} \) |
| 3 | \( 1 - 1.33T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 13 | \( 1 + 1.90T + 13T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 + 0.521T + 29T^{2} \) |
| 31 | \( 1 - 7.67T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 - 3.11T + 43T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 - 6.90T + 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 - 1.95T + 61T^{2} \) |
| 67 | \( 1 - 4.06T + 67T^{2} \) |
| 71 | \( 1 + 1.93T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 0.859T + 79T^{2} \) |
| 83 | \( 1 + 9.66T + 83T^{2} \) |
| 89 | \( 1 + 8.29T + 89T^{2} \) |
| 97 | \( 1 - 9.15T + 97T^{2} \) |
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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
−8.421538272606022586973729620752, −7.922598390444509819933251119257, −7.60325502188123684086173289917, −6.33962201278422422955138993340, −5.35751080303691515024867477482, −4.51731263937235286712716709290, −4.08629305649279154207790169950, −3.20224401553328879377634067917, −2.56664488185413299056878328003, −0.63416809452390645320721020201,
0.63416809452390645320721020201, 2.56664488185413299056878328003, 3.20224401553328879377634067917, 4.08629305649279154207790169950, 4.51731263937235286712716709290, 5.35751080303691515024867477482, 6.33962201278422422955138993340, 7.60325502188123684086173289917, 7.922598390444509819933251119257, 8.421538272606022586973729620752
