| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s + 6·13-s − 16-s − 2·17-s + 4·19-s + 2·20-s − 25-s − 6·26-s − 2·29-s − 8·31-s − 5·32-s + 2·34-s + 6·37-s − 4·38-s − 6·40-s − 10·41-s + 4·43-s − 8·47-s + 50-s − 6·52-s − 6·53-s + 2·58-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s + 1.66·13-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.447·20-s − 1/5·25-s − 1.17·26-s − 0.371·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.648·38-s − 0.948·40-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 0.141·50-s − 0.832·52-s − 0.824·53-s + 0.262·58-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5202678593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5202678593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−14.56131372224781, −13.77543966005377, −13.40589777411292, −13.08929312596068, −12.38987600255272, −11.71019928425639, −11.25621296802717, −10.91072466153726, −10.33932561259081, −9.641480815564960, −9.212935787539554, −8.672260531617918, −8.290144152715041, −7.643088441518072, −7.400578463057217, −6.566607739992595, −5.949006217620397, −5.291314845092724, −4.640146517914150, −3.975555357988142, −3.626019400819014, −2.958083597592568, −1.707369402981610, −1.309771584613775, −0.3089492638198877,
0.3089492638198877, 1.309771584613775, 1.707369402981610, 2.958083597592568, 3.626019400819014, 3.975555357988142, 4.640146517914150, 5.291314845092724, 5.949006217620397, 6.566607739992595, 7.400578463057217, 7.643088441518072, 8.290144152715041, 8.672260531617918, 9.212935787539554, 9.641480815564960, 10.33932561259081, 10.91072466153726, 11.25621296802717, 11.71019928425639, 12.38987600255272, 13.08929312596068, 13.40589777411292, 13.77543966005377, 14.56131372224781
