| L(s) = 1 | − 2-s − 0.374·3-s + 4-s + 3.12·5-s + 0.374·6-s + 0.314·7-s − 8-s − 2.85·9-s − 3.12·10-s − 0.483·11-s − 0.374·12-s + 4.55·13-s − 0.314·14-s − 1.17·15-s + 16-s + 0.582·17-s + 2.85·18-s − 7.87·19-s + 3.12·20-s − 0.117·21-s + 0.483·22-s + 2.45·23-s + 0.374·24-s + 4.76·25-s − 4.55·26-s + 2.19·27-s + 0.314·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.216·3-s + 0.5·4-s + 1.39·5-s + 0.152·6-s + 0.118·7-s − 0.353·8-s − 0.953·9-s − 0.987·10-s − 0.145·11-s − 0.108·12-s + 1.26·13-s − 0.0840·14-s − 0.302·15-s + 0.250·16-s + 0.141·17-s + 0.674·18-s − 1.80·19-s + 0.698·20-s − 0.0257·21-s + 0.103·22-s + 0.512·23-s + 0.0764·24-s + 0.952·25-s − 0.894·26-s + 0.422·27-s + 0.0594·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.583260817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.583260817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 0.374T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 - 0.314T + 7T^{2} \) |
| 11 | \( 1 + 0.483T + 11T^{2} \) |
| 13 | \( 1 - 4.55T + 13T^{2} \) |
| 17 | \( 1 - 0.582T + 17T^{2} \) |
| 19 | \( 1 + 7.87T + 19T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 - 8.31T + 29T^{2} \) |
| 31 | \( 1 + 5.53T + 31T^{2} \) |
| 37 | \( 1 - 4.13T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 - 1.24T + 53T^{2} \) |
| 59 | \( 1 - 8.47T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 7.25T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 9.13T + 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 - 4.59T + 89T^{2} \) |
| 97 | \( 1 + 2.96T + 97T^{2} \) |
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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
−8.492313348053011771566770308512, −8.070317436454844666507707741845, −6.57215770824687006503258055235, −6.46289179828982729515093308597, −5.66859050267265077754836140633, −4.95272541603756353156703480834, −3.68300123847371448804892883017, −2.60599198972376042758197701367, −1.93639925137896585897234297989, −0.815023314556506321617799790519,
0.815023314556506321617799790519, 1.93639925137896585897234297989, 2.60599198972376042758197701367, 3.68300123847371448804892883017, 4.95272541603756353156703480834, 5.66859050267265077754836140633, 6.46289179828982729515093308597, 6.57215770824687006503258055235, 8.070317436454844666507707741845, 8.492313348053011771566770308512
