L(s) = 1 | + (0.900 − 1.56i)2-s + (−0.623 − 1.07i)3-s + (−1.12 − 1.94i)4-s − 2.24·6-s − 2.24·8-s + (−0.277 + 0.480i)9-s + (−1.40 + 2.42i)12-s − 1.80·13-s + (−0.900 + 1.56i)16-s + (0.222 + 0.385i)17-s + (0.5 + 0.866i)18-s + (0.222 − 0.385i)19-s + (−0.623 + 1.07i)23-s + (1.40 + 2.42i)24-s + (−0.5 − 0.866i)25-s + (−1.62 + 2.81i)26-s + ⋯ |
L(s) = 1 | + (0.900 − 1.56i)2-s + (−0.623 − 1.07i)3-s + (−1.12 − 1.94i)4-s − 2.24·6-s − 2.24·8-s + (−0.277 + 0.480i)9-s + (−1.40 + 2.42i)12-s − 1.80·13-s + (−0.900 + 1.56i)16-s + (0.222 + 0.385i)17-s + (0.5 + 0.866i)18-s + (0.222 − 0.385i)19-s + (−0.623 + 1.07i)23-s + (1.40 + 2.42i)24-s + (−0.5 − 0.866i)25-s + (−1.62 + 2.81i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.002679729\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.002679729\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−9.164365963390190177880734472087, −7.75094982390089087475836397043, −7.17652178202595137530379775050, −6.01661406167236716670659082192, −5.48795262029539389189878875490, −4.56913344856577448702948722213, −3.69485738022074715154526430086, −2.46986856091967004084412632515, −1.89413076196815578054865504404, −0.55953660300118954990355198473,
2.71033009321280303841332942819, 3.88736094479582720847191612811, 4.62354146552116536650053120354, 5.06076217871056130586479215509, 5.81953676659684135753179232953, 6.55812162235693510804285214602, 7.56431462177653868759066503044, 7.903282802228966395340160906042, 9.158477423166957098957490893041, 9.745594627877668701278594924418