| L(s) = 1 | − 2-s + 4-s + 2.82·5-s − 8-s − 2.82·10-s + 2·11-s + 16-s − 1.41·17-s + 7.07·19-s + 2.82·20-s − 2·22-s + 4·23-s + 3.00·25-s − 2·29-s − 8.48·31-s − 32-s + 1.41·34-s + 10·37-s − 7.07·38-s − 2.82·40-s − 9.89·41-s + 2·43-s + 2·44-s − 4·46-s + 2.82·47-s − 3.00·50-s + 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.26·5-s − 0.353·8-s − 0.894·10-s + 0.603·11-s + 0.250·16-s − 0.342·17-s + 1.62·19-s + 0.632·20-s − 0.426·22-s + 0.834·23-s + 0.600·25-s − 0.371·29-s − 1.52·31-s − 0.176·32-s + 0.242·34-s + 1.64·37-s − 1.14·38-s − 0.447·40-s − 1.54·41-s + 0.304·43-s + 0.301·44-s − 0.589·46-s + 0.412·47-s − 0.424·50-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510461815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510461815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−9.769944579881218004385660802753, −9.495341183895255823792238557222, −8.714418903789153403984824790075, −7.55772417000874592739120469918, −6.78307803372587138832112294560, −5.87224703261557223156336786358, −5.09921554447328176462041806446, −3.52686130493695088434950481777, −2.29850546468802787455292500548, −1.20638853268741970222433572774,
1.20638853268741970222433572774, 2.29850546468802787455292500548, 3.52686130493695088434950481777, 5.09921554447328176462041806446, 5.87224703261557223156336786358, 6.78307803372587138832112294560, 7.55772417000874592739120469918, 8.714418903789153403984824790075, 9.495341183895255823792238557222, 9.769944579881218004385660802753
