| L(s) = 1 | − 2.82·5-s − 0.828·7-s − 0.828·11-s + 13-s − 4·17-s − 0.828·19-s + 4·23-s + 3.00·25-s − 4·29-s + 10.4·31-s + 2.34·35-s + 2·37-s − 1.17·41-s + 5.65·43-s + 6.48·47-s − 6.31·49-s − 2.34·53-s + 2.34·55-s + 0.828·59-s + 9.31·61-s − 2.82·65-s + 0.828·67-s + 14.4·71-s + 6·73-s + 0.686·77-s + 4·79-s − 8.82·83-s + ⋯ |
| L(s) = 1 | − 1.26·5-s − 0.313·7-s − 0.249·11-s + 0.277·13-s − 0.970·17-s − 0.190·19-s + 0.834·23-s + 0.600·25-s − 0.742·29-s + 1.88·31-s + 0.396·35-s + 0.328·37-s − 0.182·41-s + 0.862·43-s + 0.945·47-s − 0.901·49-s − 0.321·53-s + 0.315·55-s + 0.107·59-s + 1.19·61-s − 0.350·65-s + 0.101·67-s + 1.71·71-s + 0.702·73-s + 0.0782·77-s + 0.450·79-s − 0.969·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.068027690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068027690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 - 0.828T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 0.828T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 8.82T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−9.103908612675923452079127297892, −8.378029854677077491711979960021, −7.73011920376894555226821416921, −6.91482545291643159379085703303, −6.19315682071802761847884512389, −4.99862145026304318003221416791, −4.22702497616863086091706436193, −3.43449583228082154987019949962, −2.39409239714770356407675075129, −0.68307113866169886014610049280,
0.68307113866169886014610049280, 2.39409239714770356407675075129, 3.43449583228082154987019949962, 4.22702497616863086091706436193, 4.99862145026304318003221416791, 6.19315682071802761847884512389, 6.91482545291643159379085703303, 7.73011920376894555226821416921, 8.378029854677077491711979960021, 9.103908612675923452079127297892
