| L(s) = 1 | + 1.20·2-s − 3.41·3-s − 0.546·4-s + 1.06·5-s − 4.11·6-s − 0.0877·7-s − 3.07·8-s + 8.62·9-s + 1.28·10-s + 11-s + 1.86·12-s + 4.90·13-s − 0.105·14-s − 3.62·15-s − 2.60·16-s + 1.41·17-s + 10.4·18-s − 6.65·19-s − 0.580·20-s + 0.299·21-s + 1.20·22-s − 7.89·23-s + 10.4·24-s − 3.87·25-s + 5.90·26-s − 19.1·27-s + 0.0479·28-s + ⋯ |
| L(s) = 1 | + 0.852·2-s − 1.96·3-s − 0.273·4-s + 0.474·5-s − 1.67·6-s − 0.0331·7-s − 1.08·8-s + 2.87·9-s + 0.404·10-s + 0.301·11-s + 0.538·12-s + 1.35·13-s − 0.0282·14-s − 0.935·15-s − 0.651·16-s + 0.342·17-s + 2.45·18-s − 1.52·19-s − 0.129·20-s + 0.0652·21-s + 0.257·22-s − 1.64·23-s + 2.13·24-s − 0.774·25-s + 1.15·26-s − 3.69·27-s + 0.00906·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 3 | \( 1 + 3.41T + 3T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 + 0.0877T + 7T^{2} \) |
| 13 | \( 1 - 4.90T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 6.65T + 19T^{2} \) |
| 23 | \( 1 + 7.89T + 23T^{2} \) |
| 29 | \( 1 + 6.62T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 - 2.53T + 37T^{2} \) |
| 41 | \( 1 - 6.68T + 41T^{2} \) |
| 43 | \( 1 - 6.15T + 43T^{2} \) |
| 47 | \( 1 - 6.91T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 6.64T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 + 3.05T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 83 | \( 1 + 6.99T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 5.35T + 97T^{2} \) |
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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
−9.921724627050682310395869193641, −9.183499398138609539880646886268, −7.82518217893950477367980878328, −6.49775451781669388147569087222, −5.90336367194181683635968823464, −5.65062536455389334683384058449, −4.30832736123435744653572657263, −3.93502119830725419093302470385, −1.68487203423684173531174812624, 0,
1.68487203423684173531174812624, 3.93502119830725419093302470385, 4.30832736123435744653572657263, 5.65062536455389334683384058449, 5.90336367194181683635968823464, 6.49775451781669388147569087222, 7.82518217893950477367980878328, 9.183499398138609539880646886268, 9.921724627050682310395869193641
