L(s) = 1 | + (0.665 + 1.30i)2-s + (−0.130 + 0.822i)3-s + (−0.0875 + 0.120i)4-s + (−1.11 − 1.93i)5-s + (−1.16 + 0.377i)6-s + (−4.16 + 0.659i)7-s + (2.68 + 0.424i)8-s + (2.19 + 0.712i)9-s + (1.78 − 2.74i)10-s + (−0.920 − 3.18i)11-s + (−0.0877 − 0.0877i)12-s + (−0.824 + 0.420i)13-s + (−3.63 − 4.99i)14-s + (1.73 − 0.667i)15-s + (1.32 + 4.06i)16-s + (1.71 + 0.875i)17-s + ⋯ |
L(s) = 1 | + (0.470 + 0.923i)2-s + (−0.0751 + 0.474i)3-s + (−0.0437 + 0.0602i)4-s + (−0.500 − 0.865i)5-s + (−0.473 + 0.153i)6-s + (−1.57 + 0.249i)7-s + (0.947 + 0.150i)8-s + (0.731 + 0.237i)9-s + (0.564 − 0.869i)10-s + (−0.277 − 0.960i)11-s + (−0.0253 − 0.0253i)12-s + (−0.228 + 0.116i)13-s + (−0.970 − 1.33i)14-s + (0.448 − 0.172i)15-s + (0.330 + 1.01i)16-s + (0.416 + 0.212i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.847753 + 0.475623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.847753 + 0.475623i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.11 + 1.93i)T \) |
| 11 | \( 1 + (0.920 + 3.18i)T \) |
good | 2 | \( 1 + (-0.665 - 1.30i)T + (-1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (0.130 - 0.822i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (4.16 - 0.659i)T + (6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (0.824 - 0.420i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.71 - 0.875i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (4.39 - 3.19i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.95 + 1.95i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.810 - 0.588i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.131 + 0.403i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.771 + 4.87i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.339 - 0.467i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.05 - 5.05i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.17 + 0.186i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (4.12 + 8.09i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-5.47 + 7.53i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.40 + 2.40i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (3.05 + 3.05i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.65 - 8.17i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.854 + 5.39i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (0.705 - 2.17i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.902 + 1.77i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 - 13.9iT - 89T^{2} \) |
| 97 | \( 1 + (2.96 - 1.50i)T + (57.0 - 78.4i)T^{2} \) |
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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
−15.88869466989159642641162020774, −14.70078337587029487821289739765, −13.22185345116672715294314000912, −12.59250520005231834801654732154, −10.77042267415301300108535800056, −9.593246161391771818949980476118, −8.130736577956907968351049969863, −6.63552616113035451628106542794, −5.42197055271681493567331948377, −3.94909365557560216297173276784,
2.74666843717127578359086496276, 4.10568300404509122007929224808, 6.73196355452496030552024894922, 7.39813310497359842794180088256, 9.801561978569717485015801067635, 10.60016570277762232262630911746, 12.02294359702003045598072963014, 12.74063914366815483038840671149, 13.52146276927676897866131141808, 15.15922664116966947900724506422