L(s) = 1 | − 7-s − 11-s + 6·13-s + 7·17-s − 19-s + 8·23-s + 6·29-s − 4·31-s + 8·37-s + 5·41-s + 6·47-s + 49-s − 4·53-s − 4·59-s + 6·61-s + 5·67-s + 14·71-s + 15·73-s + 77-s − 14·79-s + 83-s + 3·89-s − 6·91-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.301·11-s + 1.66·13-s + 1.69·17-s − 0.229·19-s + 1.66·23-s + 1.11·29-s − 0.718·31-s + 1.31·37-s + 0.780·41-s + 0.875·47-s + 1/7·49-s − 0.549·53-s − 0.520·59-s + 0.768·61-s + 0.610·67-s + 1.66·71-s + 1.75·73-s + 0.113·77-s − 1.57·79-s + 0.109·83-s + 0.317·89-s − 0.628·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.060080361\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.060080361\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.40715962391343, −14.85288175540712, −14.18498930165538, −13.85567288657160, −13.04812321407803, −12.79634949190017, −12.26081326476000, −11.44062186337430, −10.99984149679018, −10.56574973515141, −9.872430508519591, −9.307750214357322, −8.789184135737697, −8.077473788206829, −7.721468380918417, −6.839771712653654, −6.394877575780834, −5.663408004637243, −5.269967256693525, −4.357442386663655, −3.650739327296509, −3.139752376831847, −2.439082797368875, −1.217250420814838, −0.8305227018621691,
0.8305227018621691, 1.217250420814838, 2.439082797368875, 3.139752376831847, 3.650739327296509, 4.357442386663655, 5.269967256693525, 5.663408004637243, 6.394877575780834, 6.839771712653654, 7.721468380918417, 8.077473788206829, 8.789184135737697, 9.307750214357322, 9.872430508519591, 10.56574973515141, 10.99984149679018, 11.44062186337430, 12.26081326476000, 12.79634949190017, 13.04812321407803, 13.85567288657160, 14.18498930165538, 14.85288175540712, 15.40715962391343