| L(s) = 1 | − 2·7-s + 2·11-s + 13-s + 17-s − 4·19-s + 4·23-s + 10·29-s − 6·31-s + 4·37-s + 8·41-s + 4·43-s − 12·47-s − 3·49-s + 10·53-s + 4·59-s + 2·61-s − 12·67-s − 6·71-s − 16·73-s − 4·77-s − 8·79-s − 8·83-s − 14·89-s − 2·91-s + 12·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.603·11-s + 0.277·13-s + 0.242·17-s − 0.917·19-s + 0.834·23-s + 1.85·29-s − 1.07·31-s + 0.657·37-s + 1.24·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s + 1.37·53-s + 0.520·59-s + 0.256·61-s − 1.46·67-s − 0.712·71-s − 1.87·73-s − 0.455·77-s − 0.900·79-s − 0.878·83-s − 1.48·89-s − 0.209·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.805464422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.805464422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.58611072830469, −11.97170142274271, −11.45798538221891, −11.23732442514171, −10.52320246941929, −10.16693989451783, −9.864326045473866, −9.076194805308759, −8.943369028027060, −8.497381927114058, −7.850985760891117, −7.371179299086344, −6.818862508671276, −6.516430392092322, −5.972025946861768, −5.644061255033681, −4.892639186238827, −4.334906994411998, −4.066962712854013, −3.320865581069456, −2.887484956731032, −2.458787036904924, −1.564530811926978, −1.135886964145459, −0.3643843169481761,
0.3643843169481761, 1.135886964145459, 1.564530811926978, 2.458787036904924, 2.887484956731032, 3.320865581069456, 4.066962712854013, 4.334906994411998, 4.892639186238827, 5.644061255033681, 5.972025946861768, 6.516430392092322, 6.818862508671276, 7.371179299086344, 7.850985760891117, 8.497381927114058, 8.943369028027060, 9.076194805308759, 9.864326045473866, 10.16693989451783, 10.52320246941929, 11.23732442514171, 11.45798538221891, 11.97170142274271, 12.58611072830469
