| L(s) = 1 | + 2-s − 0.00478·3-s + 4-s − 1.04·5-s − 0.00478·6-s − 7-s + 8-s − 2.99·9-s − 1.04·10-s − 4.80·11-s − 0.00478·12-s + 3.58·13-s − 14-s + 0.00497·15-s + 16-s − 6.75·17-s − 2.99·18-s − 1.04·20-s + 0.00478·21-s − 4.80·22-s + 0.904·23-s − 0.00478·24-s − 3.91·25-s + 3.58·26-s + 0.0287·27-s − 28-s + 3.14·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.00276·3-s + 0.5·4-s − 0.465·5-s − 0.00195·6-s − 0.377·7-s + 0.353·8-s − 0.999·9-s − 0.328·10-s − 1.44·11-s − 0.00138·12-s + 0.994·13-s − 0.267·14-s + 0.00128·15-s + 0.250·16-s − 1.63·17-s − 0.707·18-s − 0.232·20-s + 0.00104·21-s − 1.02·22-s + 0.188·23-s − 0.000976·24-s − 0.783·25-s + 0.703·26-s + 0.00552·27-s − 0.188·28-s + 0.584·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.892044919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892044919\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.00478T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 23 | \( 1 - 0.904T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 - 2.02T + 31T^{2} \) |
| 37 | \( 1 - 9.28T + 37T^{2} \) |
| 41 | \( 1 - 8.12T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 8.76T + 53T^{2} \) |
| 59 | \( 1 + 1.50T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 8.96T + 67T^{2} \) |
| 71 | \( 1 - 8.59T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.247T + 83T^{2} \) |
| 89 | \( 1 - 3.50T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.093167821472905385247647949635, −7.58761606467805411541439466639, −6.59493578782603453550771688016, −5.97319756204882226155096253543, −5.41460483983170797041031449854, −4.40571304726643292805871964158, −3.87371639876998392209933069876, −2.70226575666662815786919384587, −2.44919386209120052547119187738, −0.63771383293186674504646569121,
0.63771383293186674504646569121, 2.44919386209120052547119187738, 2.70226575666662815786919384587, 3.87371639876998392209933069876, 4.40571304726643292805871964158, 5.41460483983170797041031449854, 5.97319756204882226155096253543, 6.59493578782603453550771688016, 7.58761606467805411541439466639, 8.093167821472905385247647949635
