| L(s) = 1 | − 2.99i·2-s + (9.94 − 12.0i)3-s + 23.0·4-s − 61.8·5-s + (−35.9 − 29.7i)6-s + (92.4 − 90.9i)7-s − 164. i·8-s + (−45.3 − 238. i)9-s + 185. i·10-s + 615. i·11-s + (228. − 276. i)12-s + 322. i·13-s + (−272. − 276. i)14-s + (−614. + 742. i)15-s + 243.·16-s + 1.53e3·17-s + ⋯ |
| L(s) = 1 | − 0.529i·2-s + (0.637 − 0.770i)3-s + 0.719·4-s − 1.10·5-s + (−0.407 − 0.337i)6-s + (0.712 − 0.701i)7-s − 0.910i·8-s + (−0.186 − 0.982i)9-s + 0.585i·10-s + 1.53i·11-s + (0.459 − 0.554i)12-s + 0.529i·13-s + (−0.371 − 0.377i)14-s + (−0.705 + 0.851i)15-s + 0.237·16-s + 1.29·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0855 + 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0855 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.29892 - 1.19217i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29892 - 1.19217i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-9.94 + 12.0i)T \) |
| 7 | \( 1 + (-92.4 + 90.9i)T \) |
| good | 2 | \( 1 + 2.99iT - 32T^{2} \) |
| 5 | \( 1 + 61.8T + 3.12e3T^{2} \) |
| 11 | \( 1 - 615. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 322. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.53e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.17e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 793. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 513. iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 161. iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 8.59e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.76e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.34e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.33e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.37e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 9.42e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.00e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.48e4iT - 8.58e9T^{2} \) |
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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
−16.84786755306955703613410261289, −15.29751368078375117406365418119, −14.31311481171054108600760317548, −12.41138725253646985888029107728, −11.82795319654107956178420326956, −10.14909997157031681961061765149, −7.901710423339336425712311927184, −7.10367051124798337112483539758, −3.79465271022214912122947768145, −1.61138893560570523559857649571,
3.14837008572207202202926695387, 5.39831175283800354483346163180, 7.74934704305209645811751834164, 8.613030682939166301762443091446, 10.86627789770309310373498126874, 11.77386480814045390782350294190, 14.05856360413463194600655884529, 15.27833380029895034921781935486, 15.75568243370242246775373802763, 16.92250465888905708697315859353
