| L(s) = 1 | + 2-s + 0.666·3-s + 4-s + 2.46·5-s + 0.666·6-s − 4.80·7-s + 8-s − 2.55·9-s + 2.46·10-s + 2.95·11-s + 0.666·12-s + 3.44·13-s − 4.80·14-s + 1.64·15-s + 16-s + 17-s − 2.55·18-s + 4.63·19-s + 2.46·20-s − 3.20·21-s + 2.95·22-s + 9.28·23-s + 0.666·24-s + 1.05·25-s + 3.44·26-s − 3.70·27-s − 4.80·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.384·3-s + 0.5·4-s + 1.10·5-s + 0.272·6-s − 1.81·7-s + 0.353·8-s − 0.851·9-s + 0.778·10-s + 0.890·11-s + 0.192·12-s + 0.954·13-s − 1.28·14-s + 0.423·15-s + 0.250·16-s + 0.242·17-s − 0.602·18-s + 1.06·19-s + 0.550·20-s − 0.698·21-s + 0.629·22-s + 1.93·23-s + 0.136·24-s + 0.211·25-s + 0.674·26-s − 0.712·27-s − 0.907·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.411419725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.411419725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 0.666T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 - 3.44T + 13T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 - 9.28T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 - 2.86T + 31T^{2} \) |
| 37 | \( 1 - 0.378T + 37T^{2} \) |
| 41 | \( 1 - 3.02T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 - 5.49T + 47T^{2} \) |
| 53 | \( 1 - 1.93T + 53T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 + 16.0T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 3.97T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.10T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 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
−9.327519276071175063708499620079, −8.635632265090930346928011954193, −7.26232568973871611253261729289, −6.60111033015230880672551561424, −5.87555104840150091338697211893, −5.48992105819383482255500321878, −3.97673709002288715359584223828, −3.18422647105200156839141420558, −2.66195311120346644131019790914, −1.16909203558708605969281012459,
1.16909203558708605969281012459, 2.66195311120346644131019790914, 3.18422647105200156839141420558, 3.97673709002288715359584223828, 5.48992105819383482255500321878, 5.87555104840150091338697211893, 6.60111033015230880672551561424, 7.26232568973871611253261729289, 8.635632265090930346928011954193, 9.327519276071175063708499620079
