| L(s) = 1 | + 1.37·2-s − 2.08·3-s − 0.110·4-s − 2.34·5-s − 2.86·6-s + 1.29·7-s − 2.90·8-s + 1.34·9-s − 3.22·10-s − 11-s + 0.230·12-s + 1.78·14-s + 4.88·15-s − 3.76·16-s − 5.06·17-s + 1.85·18-s − 1.52·19-s + 0.258·20-s − 2.70·21-s − 1.37·22-s + 3.16·23-s + 6.04·24-s + 0.489·25-s + 3.44·27-s − 0.143·28-s − 4.07·29-s + 6.71·30-s + ⋯ |
| L(s) = 1 | + 0.971·2-s − 1.20·3-s − 0.0552·4-s − 1.04·5-s − 1.17·6-s + 0.489·7-s − 1.02·8-s + 0.448·9-s − 1.01·10-s − 0.301·11-s + 0.0664·12-s + 0.476·14-s + 1.26·15-s − 0.941·16-s − 1.22·17-s + 0.436·18-s − 0.349·19-s + 0.0578·20-s − 0.589·21-s − 0.293·22-s + 0.659·23-s + 1.23·24-s + 0.0978·25-s + 0.663·27-s − 0.0270·28-s − 0.757·29-s + 1.22·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8181432163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8181432163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.37T + 2T^{2} \) |
| 3 | \( 1 + 2.08T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 - 1.29T + 7T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 + 1.52T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + 0.984T + 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 9.99T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 7.43T + 73T^{2} \) |
| 79 | \( 1 - 3.76T + 79T^{2} \) |
| 83 | \( 1 - 2.79T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 8.07T + 97T^{2} \) |
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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
−9.070214391117117186332246071963, −8.477136939415342956127926204705, −7.45222647287508815362920300893, −6.62502066430712369500130851690, −5.80935778879727484289164411519, −5.07590835437258586689350473929, −4.43822700611064374510687534788, −3.77345606902418287232395514514, −2.50156805531756274971521760421, −0.54194478409671768178252806727,
0.54194478409671768178252806727, 2.50156805531756274971521760421, 3.77345606902418287232395514514, 4.43822700611064374510687534788, 5.07590835437258586689350473929, 5.80935778879727484289164411519, 6.62502066430712369500130851690, 7.45222647287508815362920300893, 8.477136939415342956127926204705, 9.070214391117117186332246071963
