L(s) = 1 | + (−0.366 + 1.36i)2-s + (0.448 + 1.67i)3-s + (−1.73 − i)4-s + (−0.258 − 4.99i)5-s − 2.44·6-s + (6.75 − 1.84i)7-s + (2 − 1.99i)8-s + (−2.59 + 1.50i)9-s + (6.91 + 1.47i)10-s + (7.12 − 12.3i)11-s + (0.896 − 3.34i)12-s + (−2.75 + 2.75i)13-s + (0.0509 + 9.89i)14-s + (8.23 − 2.67i)15-s + (1.99 + 3.46i)16-s + (23.9 − 6.41i)17-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.683i)2-s + (0.149 + 0.557i)3-s + (−0.433 − 0.250i)4-s + (−0.0516 − 0.998i)5-s − 0.408·6-s + (0.964 − 0.263i)7-s + (0.250 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.691 + 0.147i)10-s + (0.647 − 1.12i)11-s + (0.0747 − 0.278i)12-s + (−0.212 + 0.212i)13-s + (0.00363 + 0.707i)14-s + (0.549 − 0.178i)15-s + (0.124 + 0.216i)16-s + (1.40 − 0.377i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.53878 + 0.220908i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53878 + 0.220908i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 + (-0.448 - 1.67i)T \) |
| 5 | \( 1 + (0.258 + 4.99i)T \) |
| 7 | \( 1 + (-6.75 + 1.84i)T \) |
good | 11 | \( 1 + (-7.12 + 12.3i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (2.75 - 2.75i)T - 169iT^{2} \) |
| 17 | \( 1 + (-23.9 + 6.41i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (0.277 - 0.159i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-13.6 - 3.64i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 24.2iT - 841T^{2} \) |
| 31 | \( 1 + (7.62 - 13.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-0.611 + 2.28i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 29.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (11.7 - 11.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-21.9 + 81.7i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (19.4 + 72.6i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (31.7 + 18.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-54.2 - 93.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (66.0 - 17.6i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 20.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (15.1 + 56.5i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (44.5 - 25.7i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (82.8 - 82.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (95.6 - 55.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-68.2 - 68.2i)T + 9.40e3iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.09535110661777339873160913213, −11.21051286898302517059230771526, −10.01217566885809300263862114562, −8.942352951689860962717143007998, −8.378620332070019120142186395364, −7.30339253627967713367146589013, −5.66475249701964029174707174219, −4.92542915956382847001634129372, −3.69087905487099815084254182005, −1.11062738048683047225672954823,
1.58274061973270565968604083577, 2.83280733470467703005480406416, 4.32208873953886456421540488868, 5.88474350675149039039908854972, 7.31769422339247790228056553472, 7.933439988513615827535600310847, 9.295675399590748225318978598724, 10.25312656606532870322206059458, 11.24337324858320484224249665229, 12.01274672645810992330204452538