L(s) = 1 | + (0.587 − 0.190i)2-s + (−0.5 + 0.363i)4-s + (0.951 + 0.309i)5-s + 0.618·7-s + (−0.587 + 0.809i)8-s + 0.618·10-s + (−0.951 + 0.309i)11-s + (0.363 − 0.118i)14-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.587 + 0.190i)20-s + (−0.5 + 0.363i)22-s + (−0.951 + 0.309i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.224i)28-s + (−0.951 − 1.30i)29-s + ⋯ |
L(s) = 1 | + (0.587 − 0.190i)2-s + (−0.5 + 0.363i)4-s + (0.951 + 0.309i)5-s + 0.618·7-s + (−0.587 + 0.809i)8-s + 0.618·10-s + (−0.951 + 0.309i)11-s + (0.363 − 0.118i)14-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.587 + 0.190i)20-s + (−0.5 + 0.363i)22-s + (−0.951 + 0.309i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.224i)28-s + (−0.951 − 1.30i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.259963405\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.259963405\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.951 - 0.309i)T \) |
good | 2 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
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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
−10.81497037841400251153920725475, −9.811462822591195747989444870640, −9.261144389954184560313227687647, −7.975106745476583637766005351509, −7.47558461381834210035221137176, −5.82900670623603928278620804420, −5.39483418429512394676594629622, −4.34399642510628811767073224228, −3.10883424376430230109029503288, −2.06740151735685846216634824003,
1.50814893227143101926670875839, 3.08998362421842038756747751363, 4.44850067275298932576405426655, 5.36651034757042425334124080628, 5.75429391676744977564228600231, 6.93998659307715974545814567733, 8.210246893119073269046367947751, 8.894533032095395275817260810357, 9.993145359804615478825722990817, 10.35635573482981870447576145611