| L(s) = 1 | − 3-s − 2·4-s − 2·5-s + 3·7-s + 9-s + 2·12-s + 2·15-s + 4·16-s − 2·17-s + 7·19-s + 4·20-s − 3·21-s + 8·23-s − 25-s − 27-s − 6·28-s + 6·29-s − 5·31-s − 6·35-s − 2·36-s + 3·37-s + 4·41-s + 4·43-s − 2·45-s − 6·47-s − 4·48-s + 2·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s + 1.13·7-s + 1/3·9-s + 0.577·12-s + 0.516·15-s + 16-s − 0.485·17-s + 1.60·19-s + 0.894·20-s − 0.654·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.13·28-s + 1.11·29-s − 0.898·31-s − 1.01·35-s − 1/3·36-s + 0.493·37-s + 0.624·41-s + 0.609·43-s − 0.298·45-s − 0.875·47-s − 0.577·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.530961245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530961245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.11385172694811, −13.99024452041958, −13.21220334351657, −12.71432957084890, −12.30454686612621, −11.56085080552137, −11.38349323393965, −10.89901353901525, −10.26394837586764, −9.631463775464055, −9.038382400726265, −8.728788824706429, −7.849650422445351, −7.710928232184410, −7.178563574513992, −6.334155053096181, −5.652916387883292, −5.075472152280536, −4.627960379282999, −4.380595968788811, −3.420466314440064, −3.068706452906848, −1.845082068743069, −1.062599643931868, −0.5494716895809140,
0.5494716895809140, 1.062599643931868, 1.845082068743069, 3.068706452906848, 3.420466314440064, 4.380595968788811, 4.627960379282999, 5.075472152280536, 5.652916387883292, 6.334155053096181, 7.178563574513992, 7.710928232184410, 7.849650422445351, 8.728788824706429, 9.038382400726265, 9.631463775464055, 10.26394837586764, 10.89901353901525, 11.38349323393965, 11.56085080552137, 12.30454686612621, 12.71432957084890, 13.21220334351657, 13.99024452041958, 14.11385172694811
