| L(s) = 1 | − 3-s + 9-s + 7·13-s − 5·17-s − 2·19-s + 3·23-s − 27-s + 3·29-s − 9·31-s − 8·37-s − 7·39-s + 3·41-s + 43-s − 8·47-s + 5·51-s + 3·53-s + 2·57-s + 7·59-s + 61-s − 12·67-s − 3·69-s − 12·71-s + 4·73-s − 12·79-s + 81-s − 3·83-s − 3·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.94·13-s − 1.21·17-s − 0.458·19-s + 0.625·23-s − 0.192·27-s + 0.557·29-s − 1.61·31-s − 1.31·37-s − 1.12·39-s + 0.468·41-s + 0.152·43-s − 1.16·47-s + 0.700·51-s + 0.412·53-s + 0.264·57-s + 0.911·59-s + 0.128·61-s − 1.46·67-s − 0.361·69-s − 1.42·71-s + 0.468·73-s − 1.35·79-s + 1/9·81-s − 0.329·83-s − 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206974240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206974240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.45839515061472, −13.08092561057319, −12.79691190771390, −12.02955987068726, −11.60125594252052, −11.07316108943581, −10.71904730477124, −10.49232664609475, −9.650352963823751, −9.082377735931055, −8.605491816989906, −8.422320589933798, −7.506086498383635, −7.015406606132141, −6.510711938789422, −6.125205498583051, −5.520760281602061, −5.052809954845814, −4.274579785202107, −3.937133666493893, −3.301857257449216, −2.589587349205167, −1.682973712115025, −1.351580409748195, −0.3548087177640800,
0.3548087177640800, 1.351580409748195, 1.682973712115025, 2.589587349205167, 3.301857257449216, 3.937133666493893, 4.274579785202107, 5.052809954845814, 5.520760281602061, 6.125205498583051, 6.510711938789422, 7.015406606132141, 7.506086498383635, 8.422320589933798, 8.605491816989906, 9.082377735931055, 9.650352963823751, 10.49232664609475, 10.71904730477124, 11.07316108943581, 11.60125594252052, 12.02955987068726, 12.79691190771390, 13.08092561057319, 13.45839515061472
