| L(s) = 1 | + (−0.556 − 5.16i)3-s − 10.6i·5-s + 7.90i·7-s + (−26.3 + 5.75i)9-s − 11.7·11-s − 30.5·13-s + (−54.8 + 5.90i)15-s + 118. i·17-s − 66.4i·19-s + (40.8 − 4.40i)21-s − 166.·23-s + 12.4·25-s + (44.4 + 133. i)27-s + 111. i·29-s − 224. i·31-s + ⋯ |
| L(s) = 1 | + (−0.107 − 0.994i)3-s − 0.948i·5-s + 0.426i·7-s + (−0.977 + 0.213i)9-s − 0.322·11-s − 0.652·13-s + (−0.943 + 0.101i)15-s + 1.68i·17-s − 0.802i·19-s + (0.424 − 0.0457i)21-s − 1.50·23-s + 0.0998·25-s + (0.316 + 0.948i)27-s + 0.717i·29-s − 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2202763729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2202763729\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.556 + 5.16i)T \) |
| good | 5 | \( 1 + 10.6iT - 125T^{2} \) |
| 7 | \( 1 - 7.90iT - 343T^{2} \) |
| 11 | \( 1 + 11.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 66.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 111. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 224. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 70.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 247. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 98.7iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 189.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 529. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 811.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 833.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 2.83iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 796.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 875.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 656. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 237.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 868. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 755.T + 9.12e5T^{2} \) |
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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
−11.40704969863678068522675320148, −10.26723267087605408091281999908, −9.075997159354433876719722302277, −8.312918179285068186369453903728, −7.57281971976812154939117116461, −6.28805626993268058223653688711, −5.52660874144397691330464786860, −4.33731963899892994830238722964, −2.58645275717114236662589493902, −1.41816303195865044040707372062,
0.07443933163432204232984421554, 2.51429684114665325988059096881, 3.53768867842721890065056324818, 4.67580997673093623042036390831, 5.71804262948385557933551725610, 6.89860010500189236415631357491, 7.80363573096104533857203721892, 9.033242856937740753179787108647, 10.18870811016160112224942136202, 10.28657839375870579199926835761
