| L(s) = 1 | + (1.61 + 7.83i)2-s + (−11.0 − 11.0i)3-s + (−58.7 + 25.2i)4-s + (−15.5 − 15.5i)5-s + (68.5 − 104. i)6-s + 10.2·7-s + (−292. − 419. i)8-s + 242. i·9-s + (96.8 − 147. i)10-s + (1.30e3 − 1.30e3i)11-s + (926. + 369. i)12-s + (1.99e3 − 1.99e3i)13-s + (16.6 + 80.7i)14-s + 343. i·15-s + (2.81e3 − 2.97e3i)16-s − 1.80e3·17-s + ⋯ |
| L(s) = 1 | + (0.201 + 0.979i)2-s + (−0.408 − 0.408i)3-s + (−0.918 + 0.395i)4-s + (−0.124 − 0.124i)5-s + (0.317 − 0.482i)6-s + 0.0300·7-s + (−0.572 − 0.820i)8-s + 0.333i·9-s + (0.0968 − 0.147i)10-s + (0.984 − 0.984i)11-s + (0.536 + 0.213i)12-s + (0.910 − 0.910i)13-s + (0.00605 + 0.0294i)14-s + 0.101i·15-s + (0.687 − 0.725i)16-s − 0.366·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.30548 - 0.241535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30548 - 0.241535i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 - 7.83i)T \) |
| 3 | \( 1 + (11.0 + 11.0i)T \) |
| good | 5 | \( 1 + (15.5 + 15.5i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 10.2T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.30e3 + 1.30e3i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (-1.99e3 + 1.99e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 1.80e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-1.70e3 - 1.70e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 5.64e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.17e4 - 1.17e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 3.22e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (4.95e4 + 4.95e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 8.73e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-7.12e3 + 7.12e3i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 7.94e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-5.80e4 - 5.80e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (-9.26e3 + 9.26e3i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-3.12e4 + 3.12e4i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (-1.40e5 - 1.40e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 4.45e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 6.74e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 1.71e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-7.09e5 - 7.09e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 5.49e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 6.75e5T + 8.32e11T^{2} \) |
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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
−14.29890963010329271009880171597, −13.37102973886270784561537825891, −12.27081568331707511001795139731, −10.90276892579334200419919763018, −9.059109414205829473178700627515, −7.988825717349600890938755551327, −6.54878917205986006644532944306, −5.53693236777177520975841740718, −3.74195549328540608804690293751, −0.66720263424393154499642078070,
1.51078735942271379326089950460, 3.64811879907824308202260493923, 4.87052460896738430235409639819, 6.64300625436461630600163053292, 8.857101368257602187652958266305, 9.819161447489270267502415843815, 11.16497190091695329878260986920, 11.84106768288309199161377843225, 13.14140069827763714561428645308, 14.35806127742638545218652725875
