| L(s) = 1 | − 2-s + 4-s + i·5-s − 1.67i·7-s − 8-s − i·10-s + (2.25 + 2.43i)11-s − 2.95i·13-s + 1.67i·14-s + 16-s + 0.828·17-s − 2.26i·19-s + i·20-s + (−2.25 − 2.43i)22-s + 0.720i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447i·5-s − 0.632i·7-s − 0.353·8-s − 0.316i·10-s + (0.678 + 0.734i)11-s − 0.819i·13-s + 0.447i·14-s + 0.250·16-s + 0.200·17-s − 0.518i·19-s + 0.223i·20-s + (−0.480 − 0.519i)22-s + 0.150i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17777 - 0.123619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17777 - 0.123619i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 + (-2.25 - 2.43i)T \) |
| good | 7 | \( 1 + 1.67iT - 7T^{2} \) |
| 13 | \( 1 + 2.95iT - 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.26iT - 19T^{2} \) |
| 23 | \( 1 - 0.720iT - 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 - 4.36T + 31T^{2} \) |
| 37 | \( 1 - 4.69T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 + 5.08iT - 43T^{2} \) |
| 47 | \( 1 + 0.911iT - 47T^{2} \) |
| 53 | \( 1 + 3.71iT - 53T^{2} \) |
| 59 | \( 1 - 2.69iT - 59T^{2} \) |
| 61 | \( 1 - 8.74iT - 61T^{2} \) |
| 67 | \( 1 - 9.52T + 67T^{2} \) |
| 71 | \( 1 - 4.93iT - 71T^{2} \) |
| 73 | \( 1 + 6.36iT - 73T^{2} \) |
| 79 | \( 1 - 2.46iT - 79T^{2} \) |
| 83 | \( 1 - 8.85T + 83T^{2} \) |
| 89 | \( 1 + 7.84iT - 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−10.05479041889350996619595711669, −9.222337474597384034115657826187, −8.269855050199098254260972096690, −7.41427897254376522815932643601, −6.83705107803337846146619308688, −5.88268787245037051463225418828, −4.62092853677126427150565252560, −3.52076391008479936664220320579, −2.36520577366223383751054961149, −0.902693357665715264627831594843,
1.06783812362714290096090078442, 2.34362438418320770562685414790, 3.62801298228091702881509133188, 4.79534150361546660449232842774, 5.98163914456072388839402144006, 6.53627194577047402017788060981, 7.75544328902837023644801193553, 8.485787058703879294404583929847, 9.156371169053805923228651042827, 9.768106991128600612816903441741
