L(s) = 1 | + (4.30 + 6.74i)2-s − 15.5·3-s + (−26.8 + 58.0i)4-s + (−108. + 61.3i)5-s + (−67.1 − 105. i)6-s + 118.·7-s + (−507. + 69.1i)8-s + 243·9-s + (−882. − 469. i)10-s − 184. i·11-s + (418. − 905. i)12-s − 2.86e3i·13-s + (508. + 796. i)14-s + (1.69e3 − 956. i)15-s + (−2.65e3 − 3.12e3i)16-s − 4.93e3i·17-s + ⋯ |
L(s) = 1 | + (0.538 + 0.842i)2-s − 0.577·3-s + (−0.419 + 0.907i)4-s + (−0.871 + 0.490i)5-s + (−0.310 − 0.486i)6-s + 0.344·7-s + (−0.990 + 0.135i)8-s + 0.333·9-s + (−0.882 − 0.469i)10-s − 0.138i·11-s + (0.242 − 0.523i)12-s − 1.30i·13-s + (0.185 + 0.290i)14-s + (0.502 − 0.283i)15-s + (−0.647 − 0.762i)16-s − 1.00i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0798 + 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0798 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.123906 - 0.114382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.123906 - 0.114382i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-4.30 - 6.74i)T \) |
| 3 | \( 1 + 15.5T \) |
| 5 | \( 1 + (108. - 61.3i)T \) |
good | 7 | \( 1 - 118.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 184. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.86e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 4.93e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 4.72e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.95e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.24e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 1.58e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 7.83e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.10e5T + 4.75e9T^{2} \) |
| 43 | \( 1 - 3.91e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.35e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.54e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.06e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.91e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 5.37e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + 6.71e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.42e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 6.70e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 2.73e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 7.21e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.49e6iT - 8.32e11T^{2} \) |
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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
−13.77147877105034256785543756353, −12.38245372291288941897922993242, −11.63721206756257442941183099178, −10.24418993632171283814528168583, −8.289716456148196772761431458185, −7.42501181267824552336481009335, −6.08037161593168601534620771542, −4.76545327800234198299613213568, −3.29902051083158504282740480186, −0.06320209741952192741743995209,
1.66798372816148865764212830159, 3.92402724512754351187629670009, 4.87456137561398695056419011618, 6.47808960929486825625650080734, 8.325700419923066648319353955145, 9.736455512912034162665036222847, 11.11228450845409234752360001450, 11.81172933717606005635326664196, 12.67150727591951136557873104679, 13.92348068635600422491761261811