| L(s) = 1 | + (−1.5 + 2.59i)3-s + (−7.86 − 13.6i)5-s + (−11.2 + 14.7i)7-s + (−4.5 − 7.79i)9-s + (29.3 − 50.8i)11-s − 20.7·13-s + 47.2·15-s + (−21.4 + 37.1i)17-s + (68.5 + 118. i)19-s + (−21.3 − 51.2i)21-s + (14.5 + 25.1i)23-s + (−61.3 + 106. i)25-s + 27·27-s + 8.68·29-s + (−101. + 175. i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.703 − 1.21i)5-s + (−0.606 + 0.794i)7-s + (−0.166 − 0.288i)9-s + (0.804 − 1.39i)11-s − 0.442·13-s + 0.812·15-s + (−0.306 + 0.530i)17-s + (0.828 + 1.43i)19-s + (−0.222 − 0.532i)21-s + (0.131 + 0.228i)23-s + (−0.490 + 0.849i)25-s + 0.192·27-s + 0.0555·29-s + (−0.586 + 1.01i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9340884920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9340884920\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 2.59i)T \) |
| 7 | \( 1 + (11.2 - 14.7i)T \) |
| good | 5 | \( 1 + (7.86 + 13.6i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-29.3 + 50.8i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 20.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + (21.4 - 37.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-68.5 - 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-14.5 - 25.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 8.68T + 2.43e4T^{2} \) |
| 31 | \( 1 + (101. - 175. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-7.63 - 13.2i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-294. - 509. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (202. - 350. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-5.43 + 9.41i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-447. - 774. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (351. - 609. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-558. + 966. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-69.1 - 119. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 894.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-340. - 590. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 246.T + 9.12e5T^{2} \) |
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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
−11.55580351472438132351336863428, −10.40405258269336513397124144011, −9.149701012821729817449325908462, −8.831285144141922478383685162727, −7.74143362691639776649637758164, −6.14016735382304121644911378188, −5.46723746567584385003366608783, −4.21200242733086041385045479409, −3.25845058854128847944490249613, −1.09002819062328945496335881142,
0.41688871825703501921333026228, 2.37703251172649661290432936727, 3.63602427916605525812712353273, 4.77849484874874461495660156954, 6.49501664371190834424206350001, 7.17317984177386752241886568779, 7.47296837100554487715153835982, 9.249130290930858845773908980515, 10.06610753372694567389751797196, 11.09753253245315392394880011874
