L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (0.358 − 0.621i)5-s + (2.62 − 0.358i)7-s − 0.999i·8-s + (−0.621 + 0.358i)10-s + (2.59 − 1.5i)11-s − 2.44i·13-s + (−2.44 − i)14-s + (−0.5 + 0.866i)16-s + (2.95 + 5.12i)17-s + (−5.12 − 2.95i)19-s + 0.717·20-s − 3·22-s + (−3.67 − 2.12i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.160 − 0.277i)5-s + (0.990 − 0.135i)7-s − 0.353i·8-s + (−0.196 + 0.113i)10-s + (0.783 − 0.452i)11-s − 0.679i·13-s + (−0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.717 + 1.24i)17-s + (−1.17 − 0.678i)19-s + 0.160·20-s − 0.639·22-s + (−0.766 − 0.442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.868743 - 0.271700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.868743 - 0.271700i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.358i)T \) |
good | 5 | \( 1 + (-0.358 + 0.621i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (-2.95 - 5.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.12 + 2.95i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.67 + 2.12i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.24iT - 29T^{2} \) |
| 31 | \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.121 + 0.210i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 0.242T + 43T^{2} \) |
| 47 | \( 1 + (-2.95 + 5.12i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.27 + 3.62i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.03 + 6.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.878 - 0.507i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.75iT - 71T^{2} \) |
| 73 | \( 1 + (1.24 - 0.717i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−13.04919587853276655503885343637, −12.17961734461077163123430508229, −11.00889591995896199800984117752, −10.36097866751794932791629908042, −8.842859915429263761776500206928, −8.286696477906000483278812320345, −6.87941292926266907766017284983, −5.33816727758436827108866961852, −3.70555517225141664449845033932, −1.60834260819349869734916631043,
1.96378941236481054100875427111, 4.33296126047713043764598076727, 5.84533253371431099379475615897, 7.07772438518680706746806924718, 8.116893152052058923129282897091, 9.242140803145329833134425611150, 10.20827555350745823817352543257, 11.42206667533199962951889883363, 12.13388182003623915313219001650, 13.86607376711173754908863808815