L(s) = 1 | + (0.114 + 1.40i)2-s + (0.935 − 0.186i)3-s + (−1.97 + 0.323i)4-s + (0.763 + 0.510i)5-s + (0.369 + 1.29i)6-s + (−0.225 − 0.337i)7-s + (−0.681 − 2.74i)8-s + (−1.93 + 0.799i)9-s + (−0.631 + 1.13i)10-s + (0.892 − 4.48i)11-s + (−1.78 + 0.669i)12-s + (−1.21 + 1.21i)13-s + (0.450 − 0.357i)14-s + (0.809 + 0.335i)15-s + (3.79 − 1.27i)16-s + (2.80 − 3.02i)17-s + ⋯ |
L(s) = 1 | + (0.0810 + 0.996i)2-s + (0.540 − 0.107i)3-s + (−0.986 + 0.161i)4-s + (0.341 + 0.228i)5-s + (0.150 + 0.529i)6-s + (−0.0853 − 0.127i)7-s + (−0.240 − 0.970i)8-s + (−0.643 + 0.266i)9-s + (−0.199 + 0.358i)10-s + (0.269 − 1.35i)11-s + (−0.515 + 0.193i)12-s + (−0.337 + 0.337i)13-s + (0.120 − 0.0954i)14-s + (0.208 + 0.0865i)15-s + (0.947 − 0.318i)16-s + (0.679 − 0.733i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.842099 + 0.530953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.842099 + 0.530953i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.114 - 1.40i)T \) |
| 17 | \( 1 + (-2.80 + 3.02i)T \) |
good | 3 | \( 1 + (-0.935 + 0.186i)T + (2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (-0.763 - 0.510i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.225 + 0.337i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.892 + 4.48i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (1.21 - 1.21i)T - 13iT^{2} \) |
| 19 | \( 1 + (1.29 - 3.11i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (1.22 + 0.243i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (3.14 - 4.71i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (-1.67 - 8.41i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (-1.30 - 6.56i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.11 + 0.743i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-2.03 - 4.91i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (9.03 + 9.03i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.36 - 0.977i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-11.1 + 4.62i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.41 - 3.61i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + 8.38T + 67T^{2} \) |
| 71 | \( 1 + (-13.3 + 2.65i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (5.16 + 3.45i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-1.81 + 9.10i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (3.70 + 1.53i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.88 - 6.88i)T + 89iT^{2} \) |
| 97 | \( 1 + (-4.91 + 7.35i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−14.63521620429304743167236680940, −14.15460711901045428324983280189, −13.34780028083415400966926256214, −11.82025267760648611552139687417, −10.18010304019018919053999407624, −8.878942990780654076118401090948, −8.022726996497575049051345859315, −6.58328113844902444762539848536, −5.36231649901853045393200142944, −3.35594995194368132027147786507,
2.33104359564218407443710408860, 4.05640710785226018877560889382, 5.68789419177026368808061643579, 7.87365722099354254931564070283, 9.252507985724368687989510387889, 9.863441971899067346844110000688, 11.33586445183725800432899397970, 12.44336808633352561116496379391, 13.30061503347065566246961331827, 14.54445929196532535768486467316