| L(s) = 1 | + (0.238 − 1.39i)2-s + (0.0741 − 0.372i)3-s + (−1.88 − 0.664i)4-s + (−1.26 − 1.89i)5-s + (−0.501 − 0.192i)6-s + (3.23 + 2.15i)7-s + (−1.37 + 2.47i)8-s + (2.63 + 1.09i)9-s + (−2.93 + 1.31i)10-s + (−1.93 + 0.385i)11-s + (−0.387 + 0.653i)12-s + (−1.26 − 1.26i)13-s + (3.77 − 3.98i)14-s + (−0.798 + 0.330i)15-s + (3.11 + 2.50i)16-s + (−2.29 + 3.42i)17-s + ⋯ |
| L(s) = 1 | + (0.168 − 0.985i)2-s + (0.0427 − 0.215i)3-s + (−0.943 − 0.332i)4-s + (−0.565 − 0.846i)5-s + (−0.204 − 0.0784i)6-s + (1.22 + 0.815i)7-s + (−0.486 + 0.873i)8-s + (0.879 + 0.364i)9-s + (−0.929 + 0.414i)10-s + (−0.584 + 0.116i)11-s + (−0.111 + 0.188i)12-s + (−0.349 − 0.349i)13-s + (1.01 − 1.06i)14-s + (−0.206 + 0.0854i)15-s + (0.778 + 0.627i)16-s + (−0.555 + 0.831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0432 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0432 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.667145 - 0.638894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.667145 - 0.638894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.238 + 1.39i)T \) |
| 17 | \( 1 + (2.29 - 3.42i)T \) |
| good | 3 | \( 1 + (-0.0741 + 0.372i)T + (-2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (1.26 + 1.89i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-3.23 - 2.15i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (1.93 - 0.385i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (1.26 + 1.26i)T + 13iT^{2} \) |
| 19 | \( 1 + (-1.34 - 3.25i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.09 + 5.49i)T + (-21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (2.30 - 1.53i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (8.35 + 1.66i)T + (28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (0.599 + 0.119i)T + (34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (1.13 - 1.69i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (3.47 - 8.38i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-3.89 + 3.89i)T - 47iT^{2} \) |
| 53 | \( 1 + (-10.3 + 4.30i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (1.67 + 0.694i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (7.37 + 4.92i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + (1.76 - 8.86i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (3.00 + 4.49i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.434 - 0.0865i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (11.0 - 4.55i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (0.569 - 0.569i)T - 89iT^{2} \) |
| 97 | \( 1 + (-13.4 + 8.97i)T + (37.1 - 89.6i)T^{2} \) |
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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
−14.44483808703193896516075839439, −12.91052099571900454989366553830, −12.46578736733864050578938915523, −11.33634940854108310351993362489, −10.21426526042387271045266080663, −8.670453249946102529121722933712, −7.924941126191513315237239663904, −5.33333574107970286505658830370, −4.33172287984102085956392275944, −1.92780584528967335638852635054,
3.84760092359969043442533702156, 5.07142388412012570785895228594, 7.20248683265643073111922155146, 7.44636304401466247812997908215, 9.146412541607547242079858658596, 10.55832872563925809190097473523, 11.67524477999363349173346635511, 13.30058409468075746133469201754, 14.19390435232790584638314362911, 15.17000908005663286853785449795
