L(s) = 1 | + (−2 + 2i)5-s + 4.24i·7-s + (2.82 − 2.82i)11-s + (3 + 3i)13-s + 6·17-s + (1.41 + 1.41i)19-s + 2.82i·23-s − 3i·25-s + (−4 − 4i)29-s + 4.24·31-s + (−8.48 − 8.48i)35-s + (3 − 3i)37-s + 10i·41-s + (4.24 − 4.24i)43-s + 2.82·47-s + ⋯ |
L(s) = 1 | + (−0.894 + 0.894i)5-s + 1.60i·7-s + (0.852 − 0.852i)11-s + (0.832 + 0.832i)13-s + 1.45·17-s + (0.324 + 0.324i)19-s + 0.589i·23-s − 0.600i·25-s + (−0.742 − 0.742i)29-s + 0.762·31-s + (−1.43 − 1.43i)35-s + (0.493 − 0.493i)37-s + 1.56i·41-s + (0.646 − 0.646i)43-s + 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.626391263\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.626391263\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2 - 2i)T - 5iT^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + (-2.82 + 2.82i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (-1.41 - 1.41i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (4 + 4i)T + 29iT^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 + 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (4 - 4i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (-3 - 3i)T + 61iT^{2} \) |
| 67 | \( 1 + (2.82 + 2.82i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + (11.3 + 11.3i)T + 83iT^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−9.221636185498458257202756041682, −8.412905344227373436406251453878, −7.83123019165525030910594958723, −6.89992027941922379012705046960, −5.96017808759052343894584113349, −5.67748616431684756353462742180, −4.19038354001089997481501250416, −3.44187378014585571285971055976, −2.73742679783199083848775227706, −1.36442122673214005782188696933,
0.68095069863424648489079581465, 1.33475434791271000162803324306, 3.25648033433620113852056975679, 3.97848449908377327645617499458, 4.53037231747636614780756066953, 5.48051986940857899689547142481, 6.61076662830855751230710230030, 7.42157575355835099623711553998, 7.87058978211184551367311055249, 8.648229544677480520196476942816