L(s) = 1 | + (5.40 + 5.89i)2-s − 15.5i·3-s + (−5.57 + 63.7i)4-s − 55.9·5-s + (91.9 − 84.2i)6-s + 125. i·7-s + (−406. + 311. i)8-s − 243·9-s + (−302. − 329. i)10-s + 685. i·11-s + (993. + 86.9i)12-s − 3.31e3·13-s + (−739. + 677. i)14-s + 871. i·15-s + (−4.03e3 − 711. i)16-s − 3.07e3·17-s + ⋯ |
L(s) = 1 | + (0.675 + 0.737i)2-s − 0.577i·3-s + (−0.0871 + 0.996i)4-s − 0.447·5-s + (0.425 − 0.390i)6-s + 0.365i·7-s + (−0.793 + 0.608i)8-s − 0.333·9-s + (−0.302 − 0.329i)10-s + 0.515i·11-s + (0.575 + 0.0503i)12-s − 1.50·13-s + (−0.269 + 0.247i)14-s + 0.258i·15-s + (−0.984 − 0.173i)16-s − 0.625·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0455131 + 1.04219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0455131 + 1.04219i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.40 - 5.89i)T \) |
| 3 | \( 1 + 15.5iT \) |
| 5 | \( 1 + 55.9T \) |
good | 7 | \( 1 - 125. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 685. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.31e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 3.07e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 6.08e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 6.97e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.37e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 3.75e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 6.43e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 2.74e3T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.00e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 7.29e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 9.96e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.37e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.74e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.44e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.24e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.89e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.47e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 3.55e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.48e4T + 4.96e11T^{2} \) |
| 97 | \( 1 + 9.38e5T + 8.32e11T^{2} \) |
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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
−14.59708542442551538396658879660, −13.23092201891526975437300972579, −12.35996659389117385264579911897, −11.51889916689357550157097652195, −9.478443596986609005545882029178, −7.972759873528347389247739146349, −7.17212614442494032372035976747, −5.76541215856214828722193763992, −4.33513153404102210310969557619, −2.49554679499393763858388541030,
0.32545910660387441553982753296, 2.63630337636311397037611973319, 4.11716064413313513535327059852, 5.20742726086260618440690858079, 6.97204993410699788215856884047, 8.877632388893030236600932527000, 10.10963194855209993693211971993, 11.07875135245795780138737248627, 12.07139302267130867325401337007, 13.25668610053997744574200459232