L(s) = 1 | + (1 + i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (4.32 + 2.51i)5-s + 2.44·6-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (1.80 + 6.83i)10-s + 2.92·11-s + (2.44 + 2.44i)12-s + (−1.13 + 1.13i)13-s + 3.74i·14-s + (8.37 − 2.20i)15-s − 4·16-s + (1.54 + 1.54i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.864 + 0.503i)5-s + 0.408·6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.180 + 0.683i)10-s + 0.265·11-s + (0.204 + 0.204i)12-s + (−0.0871 + 0.0871i)13-s + 0.267i·14-s + (0.558 − 0.147i)15-s − 0.250·16-s + (0.0908 + 0.0908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.36764 + 1.01705i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.36764 + 1.01705i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (-4.32 - 2.51i)T \) |
| 7 | \( 1 + (-1.87 - 1.87i)T \) |
good | 11 | \( 1 - 2.92T + 121T^{2} \) |
| 13 | \( 1 + (1.13 - 1.13i)T - 169iT^{2} \) |
| 17 | \( 1 + (-1.54 - 1.54i)T + 289iT^{2} \) |
| 19 | \( 1 - 3.35iT - 361T^{2} \) |
| 23 | \( 1 + (-7.90 + 7.90i)T - 529iT^{2} \) |
| 29 | \( 1 + 13.5iT - 841T^{2} \) |
| 31 | \( 1 - 15.7T + 961T^{2} \) |
| 37 | \( 1 + (16.8 + 16.8i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 72.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-20.3 + 20.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (52.3 + 52.3i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (40.5 - 40.5i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 117. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 45.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-57.7 - 57.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 51.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-72.3 + 72.3i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 37.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (21.3 - 21.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 100. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-52.5 - 52.5i)T + 9.40e3iT^{2} \) |
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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
−12.49497129388330151931224139168, −11.49609745462188672319473877207, −10.25176153845689822202342507920, −9.159300311934763592370034973994, −8.155386301712536607046842901891, −6.97739080529161034233833321995, −6.17952208867212590912499984636, −5.00673082250196025744122644956, −3.36065343991940842874867230240, −2.00239829246164356704608878141,
1.52241950878432744418559095692, 3.04183722672410897991439383008, 4.49094011885412783433562128268, 5.38117680416497021484803468567, 6.69488137474184101702921364504, 8.251485264899906117148053027422, 9.303289986839872025524254967258, 10.04900173106769210218662660679, 10.99840000404401805780312725396, 12.10308878917076866292477555378