| L(s) = 1 | + 2-s + 3.10·3-s + 4-s − 1.10·5-s + 3.10·6-s − 7-s + 8-s + 6.62·9-s − 1.10·10-s − 5.62·11-s + 3.10·12-s − 3.62·13-s − 14-s − 3.42·15-s + 16-s + 4.52·17-s + 6.62·18-s − 0.578·19-s − 1.10·20-s − 3.10·21-s − 5.62·22-s − 23-s + 3.10·24-s − 3.78·25-s − 3.62·26-s + 11.2·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.79·3-s + 0.5·4-s − 0.493·5-s + 1.26·6-s − 0.377·7-s + 0.353·8-s + 2.20·9-s − 0.348·10-s − 1.69·11-s + 0.895·12-s − 1.00·13-s − 0.267·14-s − 0.883·15-s + 0.250·16-s + 1.09·17-s + 1.56·18-s − 0.132·19-s − 0.246·20-s − 0.677·21-s − 1.19·22-s − 0.208·23-s + 0.633·24-s − 0.756·25-s − 0.711·26-s + 2.16·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.828507031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.828507031\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 - 4.52T + 17T^{2} \) |
| 19 | \( 1 + 0.578T + 19T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 - 3.04T + 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 - 8.35T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 6.72T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−12.02096902259967353935559782809, −10.39786949962265298099056758337, −9.873031669321519012899228073313, −8.585059706524510585085920132767, −7.71633476319745884219079949222, −7.23933723213972528496961285449, −5.45143313192278675541770474822, −4.20523909726242688787618091362, −3.12782178610052497574144837483, −2.36028859015236056913774795209,
2.36028859015236056913774795209, 3.12782178610052497574144837483, 4.20523909726242688787618091362, 5.45143313192278675541770474822, 7.23933723213972528496961285449, 7.71633476319745884219079949222, 8.585059706524510585085920132767, 9.873031669321519012899228073313, 10.39786949962265298099056758337, 12.02096902259967353935559782809
