| L(s) = 1 | − 4·11-s − 13-s − 2·17-s − 4·19-s − 8·23-s + 6·29-s + 10·37-s − 6·41-s − 4·43-s − 7·49-s + 6·53-s + 12·59-s − 10·61-s − 12·67-s − 12·71-s + 10·73-s − 8·79-s − 16·83-s − 14·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1.11·29-s + 1.64·37-s − 0.937·41-s − 0.609·43-s − 49-s + 0.824·53-s + 1.56·59-s − 1.28·61-s − 1.46·67-s − 1.42·71-s + 1.17·73-s − 0.900·79-s − 1.75·83-s − 1.48·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3601235418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3601235418\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.65349787107855, −13.38418781881005, −12.89235381884141, −12.39365611103938, −11.82302703623400, −11.47232699915952, −10.68074599653670, −10.45027072324834, −9.861031864277242, −9.546818098736108, −8.550484373447416, −8.388275425058372, −7.899244499146073, −7.266294732591277, −6.708307441477766, −6.127218028933796, −5.678905493260299, −5.031896833782484, −4.349381192856854, −4.146566255797174, −3.101439416991985, −2.645879795048699, −2.079934452345857, −1.338844733992195, −0.1845386529094404,
0.1845386529094404, 1.338844733992195, 2.079934452345857, 2.645879795048699, 3.101439416991985, 4.146566255797174, 4.349381192856854, 5.031896833782484, 5.678905493260299, 6.127218028933796, 6.708307441477766, 7.266294732591277, 7.899244499146073, 8.388275425058372, 8.550484373447416, 9.546818098736108, 9.861031864277242, 10.45027072324834, 10.68074599653670, 11.47232699915952, 11.82302703623400, 12.39365611103938, 12.89235381884141, 13.38418781881005, 13.65349787107855
