| L(s) = 1 | + (3.23 + 2.35i)2-s + (4.94 + 15.2i)4-s + (−61.7 + 44.8i)5-s + (21.9 + 67.4i)7-s + (−19.7 + 60.8i)8-s − 305.·10-s + (200. + 347. i)11-s + (−331. − 241. i)13-s + (−87.6 + 269. i)14-s + (−207. + 150. i)16-s + (−94.1 + 68.3i)17-s + (286. − 880. i)19-s + (−988. − 717. i)20-s + (−169. + 1.59e3i)22-s − 2.14e3·23-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−1.10 + 0.802i)5-s + (0.168 + 0.519i)7-s + (−0.109 + 0.336i)8-s − 0.965·10-s + (0.499 + 0.866i)11-s + (−0.544 − 0.395i)13-s + (−0.119 + 0.367i)14-s + (−0.202 + 0.146i)16-s + (−0.0789 + 0.0573i)17-s + (0.181 − 0.559i)19-s + (−0.552 − 0.401i)20-s + (−0.0744 + 0.703i)22-s − 0.843·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5776768240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5776768240\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-200. - 347. i)T \) |
| good | 5 | \( 1 + (61.7 - 44.8i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-21.9 - 67.4i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (331. + 241. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (94.1 - 68.3i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-286. + 880. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 2.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (1.41e3 + 4.36e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.74e3 + 1.99e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.00e3 + 9.23e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.41e3 - 4.36e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 7.75e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (3.96e3 - 1.22e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (2.97e3 + 2.16e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.39e4 - 4.30e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.58e4 - 1.15e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 4.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-3.02e4 + 2.19e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.92e4 + 5.91e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (3.09e4 + 2.24e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-2.07e4 + 1.51e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + 8.82e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.29e5 - 9.40e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.03123335923066322554546102331, −11.62844350291299360577185152435, −10.41933147590807829492571170969, −9.119749416328511481774485967852, −7.78302385170318290929139022753, −7.23325913944347185718563159867, −6.03552733463769166070786794923, −4.66585356685449658882611044769, −3.64205771228815972383826789984, −2.33388572705329671983341327492,
0.14500466252646428287358677803, 1.46598687831862731530006725051, 3.42947934684804075075183376812, 4.25845319554188102514427477393, 5.32939839335558947469073098565, 6.79319231788243203719052530395, 7.967922750095410456301119896921, 8.908191686524073557219164639929, 10.17604514382462435803538199765, 11.30934924323024898225790156780
