| L(s) = 1 | + (3.23 − 2.35i)2-s + (4.94 − 15.2i)4-s + (−3.66 + 5.05i)5-s + (−101. − 32.9i)7-s + (−19.7 − 60.8i)8-s + 24.9i·10-s + (−278. + 288. i)11-s + (277. + 382. i)13-s + (−405. + 131. i)14-s + (−207. − 150. i)16-s + (1.57e3 + 1.14e3i)17-s + (210. − 68.3i)19-s + (58.7 + 80.8i)20-s + (−223. + 1.58e3i)22-s − 2.63e3i·23-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.0656 + 0.0903i)5-s + (−0.782 − 0.254i)7-s + (−0.109 − 0.336i)8-s + 0.0789i·10-s + (−0.694 + 0.719i)11-s + (0.455 + 0.627i)13-s + (−0.553 + 0.179i)14-s + (−0.202 − 0.146i)16-s + (1.31 + 0.958i)17-s + (0.133 − 0.0434i)19-s + (0.0328 + 0.0451i)20-s + (−0.0982 + 0.700i)22-s − 1.03i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.912185210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.912185210\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (278. - 288. i)T \) |
| good | 5 | \( 1 + (3.66 - 5.05i)T + (-965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (101. + 32.9i)T + (1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-277. - 382. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.57e3 - 1.14e3i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-210. + 68.3i)T + (2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 2.63e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (2.05e3 - 6.33e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.62e3 + 1.18e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.06e3 - 9.44e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.81e3 + 5.58e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.81e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.41e4 + 7.86e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-8.75e3 - 1.20e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-3.61e4 - 1.17e4i)T + (5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.69e4 - 2.33e4i)T + (-2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 6.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-4.73e3 + 6.51e3i)T + (-5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.61e4 - 8.51e3i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (1.91e3 + 2.63e3i)T + (-9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.19e3 + 869. i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 5.54e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.00e5 - 7.27e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.92778034784943068186534762354, −10.64576725514440429230384376960, −10.08732224584748447567860885237, −8.895856323809303512550915369608, −7.47559460622616471926265971573, −6.45048559984321738830064826418, −5.28529011535755988657881621935, −3.98666872031183822852834337544, −2.91758931564319673363814013250, −1.35216335276832718722839441091,
0.50489386441625310454340592882, 2.74169314223546387124889695872, 3.69685189027939634490214365577, 5.34461774308200282736154013903, 5.97801425741882670224040227856, 7.34792599402967708037592473932, 8.226824774721297532676606543106, 9.451513758934916694940475946202, 10.50390353379534505489259360532, 11.71356376035414598928443393018
