| L(s) = 1 | − 1.77·2-s − 1.20·3-s + 1.16·4-s + 2.53·5-s + 2.14·6-s + 1.48·8-s − 1.54·9-s − 4.50·10-s + 4.69·11-s − 1.40·12-s + 2.97·13-s − 3.05·15-s − 4.97·16-s + 7.01·17-s + 2.74·18-s − 2.78·19-s + 2.94·20-s − 8.35·22-s + 4.04·23-s − 1.79·24-s + 1.41·25-s − 5.29·26-s + 5.48·27-s + 7.66·29-s + 5.44·30-s − 1.24·31-s + 5.87·32-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.697·3-s + 0.582·4-s + 1.13·5-s + 0.877·6-s + 0.525·8-s − 0.513·9-s − 1.42·10-s + 1.41·11-s − 0.406·12-s + 0.825·13-s − 0.789·15-s − 1.24·16-s + 1.70·17-s + 0.646·18-s − 0.638·19-s + 0.659·20-s − 1.78·22-s + 0.843·23-s − 0.366·24-s + 0.283·25-s − 1.03·26-s + 1.05·27-s + 1.42·29-s + 0.993·30-s − 0.223·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9906029414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9906029414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 3 | \( 1 + 1.20T + 3T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 - 7.01T + 17T^{2} \) |
| 19 | \( 1 + 2.78T + 19T^{2} \) |
| 23 | \( 1 - 4.04T + 23T^{2} \) |
| 29 | \( 1 - 7.66T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 + 5.48T + 37T^{2} \) |
| 43 | \( 1 - 9.18T + 43T^{2} \) |
| 47 | \( 1 - 7.77T + 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 0.292T + 83T^{2} \) |
| 89 | \( 1 + 2.99T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−9.112900033431056507069599601453, −8.699305183743703792002554061271, −7.75279937108280077258093824694, −6.70745947515470906365086879663, −6.14555107067643667636352165363, −5.43790208751979290518068398079, −4.35180591228141789415630654582, −3.04677369887514431798764488254, −1.61083231568308690084749804186, −0.928265437406178625050439177719,
0.928265437406178625050439177719, 1.61083231568308690084749804186, 3.04677369887514431798764488254, 4.35180591228141789415630654582, 5.43790208751979290518068398079, 6.14555107067643667636352165363, 6.70745947515470906365086879663, 7.75279937108280077258093824694, 8.699305183743703792002554061271, 9.112900033431056507069599601453
