L(s) = 1 | + (4.16 + 3.83i)2-s + (−7.28 + 13.7i)3-s + (2.66 + 31.8i)4-s + (−45.7 + 32.1i)5-s + (−83.1 + 29.4i)6-s + 213.·7-s + (−111. + 142. i)8-s + (−136. − 200. i)9-s + (−313. − 41.1i)10-s − 602.·11-s + (−458. − 195. i)12-s − 333. i·13-s + (887. + 816. i)14-s + (−110. − 864. i)15-s + (−1.00e3 + 169. i)16-s + 831.·17-s + ⋯ |
L(s) = 1 | + (0.735 + 0.677i)2-s + (−0.467 + 0.883i)3-s + (0.0831 + 0.996i)4-s + (−0.817 + 0.575i)5-s + (−0.942 + 0.334i)6-s + 1.64·7-s + (−0.613 + 0.789i)8-s + (−0.562 − 0.826i)9-s + (−0.991 − 0.130i)10-s − 1.50·11-s + (−0.919 − 0.392i)12-s − 0.547i·13-s + (1.21 + 1.11i)14-s + (−0.126 − 0.991i)15-s + (−0.986 + 0.165i)16-s + 0.697·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.168562 - 1.59962i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.168562 - 1.59962i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-4.16 - 3.83i)T \) |
| 3 | \( 1 + (7.28 - 13.7i)T \) |
| 5 | \( 1 + (45.7 - 32.1i)T \) |
good | 7 | \( 1 - 213.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 602.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 333. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 831.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.07e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.99e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.59e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.39e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 9.66e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 669. iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.40e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.97e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 869.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.71e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.09e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.86e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 2.98e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.69e5iT - 8.58e9T^{2} \) |
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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
−14.98498625727315981435016209839, −13.97456460843310989203159479670, −12.28004963531878891466682016688, −11.34943932174921400765986512069, −10.47124057415361945986813458938, −8.284215300192161106365677380312, −7.56164345107185689842218262519, −5.58541839841147643080093769158, −4.71470431921275169673981664382, −3.24707464633497590095928023866,
0.64351940014693375775688507419, 2.21147419088406553433847965962, 4.58152477480610802375718495401, 5.44693009925680922769882405008, 7.40875298716354982777491412814, 8.454707319157298531120670652548, 10.67459258746152032752754805666, 11.45767294040001379352721094317, 12.27616044244472444092629576816, 13.25739176340558046326619765582