L(s) = 1 | + (−0.707 − 0.707i)3-s + (1.02 + 1.02i)5-s + (−2.25 − 1.37i)7-s + 1.00i·9-s + (0.872 + 0.872i)11-s + (2.13 − 2.13i)13-s − 1.44i·15-s + 2.47i·17-s + (0.479 + 0.479i)19-s + (0.622 + 2.57i)21-s + 0.386·23-s − 2.89i·25-s + (0.707 − 0.707i)27-s + (−2.27 − 2.27i)29-s + 10.5·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.458 + 0.458i)5-s + (−0.853 − 0.520i)7-s + 0.333i·9-s + (0.263 + 0.263i)11-s + (0.591 − 0.591i)13-s − 0.374i·15-s + 0.600i·17-s + (0.110 + 0.110i)19-s + (0.135 + 0.561i)21-s + 0.0805·23-s − 0.579i·25-s + (0.136 − 0.136i)27-s + (−0.422 − 0.422i)29-s + 1.89·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.409976756\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409976756\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + (2.25 + 1.37i)T \) |
good | 5 | \( 1 + (-1.02 - 1.02i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.872 - 0.872i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.13 + 2.13i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 + (-0.479 - 0.479i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.386T + 23T^{2} \) |
| 29 | \( 1 + (2.27 + 2.27i)T + 29iT^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + (-2.46 + 2.46i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.121T + 41T^{2} \) |
| 43 | \( 1 + (6.98 + 6.98i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + (3.51 - 3.51i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.73 + 9.73i)T - 59iT^{2} \) |
| 61 | \( 1 + (-8.19 + 8.19i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.20 - 7.20i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + 8.75iT - 79T^{2} \) |
| 83 | \( 1 + (3.89 + 3.89i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.14T + 89T^{2} \) |
| 97 | \( 1 - 0.654iT - 97T^{2} \) |
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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
−9.796204623440671835436525318628, −8.643667966592299804912651911353, −7.80417885563434271877066509043, −6.83526881007734873021026614456, −6.31775293366079486301459261749, −5.60458834419065069669870998052, −4.31467125211502502436054803044, −3.34222401396003606757316903905, −2.21264839892868351589253229276, −0.75983135327990431832120610201,
1.08442264298433014709429442035, 2.62858718747621791881659946787, 3.67201961146173798453876445917, 4.72482652438257796128047665028, 5.59449107686603236327935425024, 6.30169278338100986988274905785, 7.02341113449625795235170053562, 8.371333000097662254234224659480, 9.077361441565292171957335607854, 9.631462246748802325478618622487