| L(s) = 1 | + 4.04·5-s + 3.89i·7-s − 0.468i·11-s + 4.89i·13-s + 3.57i·17-s + 1.89·19-s − 8.56·23-s + 11.3·25-s − 2.36·29-s − 7.18i·31-s + 15.7i·35-s + 6.16i·37-s + 5.13i·41-s + 10.5·43-s − 9.37·47-s + ⋯ |
| L(s) = 1 | + 1.80·5-s + 1.47i·7-s − 0.141i·11-s + 1.35i·13-s + 0.867i·17-s + 0.435·19-s − 1.78·23-s + 2.27·25-s − 0.439·29-s − 1.28i·31-s + 2.66i·35-s + 1.01i·37-s + 0.801i·41-s + 1.60·43-s − 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.517861926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.517861926\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.04T + 5T^{2} \) |
| 7 | \( 1 - 3.89iT - 7T^{2} \) |
| 11 | \( 1 + 0.468iT - 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 3.57iT - 17T^{2} \) |
| 19 | \( 1 - 1.89T + 19T^{2} \) |
| 23 | \( 1 + 8.56T + 23T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 + 7.18iT - 31T^{2} \) |
| 37 | \( 1 - 6.16iT - 37T^{2} \) |
| 41 | \( 1 - 5.13iT - 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 + 6.26iT - 59T^{2} \) |
| 61 | \( 1 - 0.983iT - 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 3.02T + 71T^{2} \) |
| 73 | \( 1 - 4.58T + 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 + 5.66iT - 83T^{2} \) |
| 89 | \( 1 + 3.17iT - 89T^{2} \) |
| 97 | \( 1 - 0.611T + 97T^{2} \) |
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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
−8.609989098316286580773905780484, −7.84039062572283370010906058233, −6.62857216624454375771172894803, −6.05599831861940467482196830875, −5.81902209905566539418508866304, −4.95821852366538626050210812841, −4.01624171375977717088799239446, −2.75914506048675529703603472971, −2.07419662854440882925794031436, −1.59693894836644210604832192680,
0.60892134528435739558340288939, 1.55899272506758709768386053933, 2.50503371342009260030923712794, 3.39300046212121243255947191513, 4.34583901354568807352504035604, 5.30136081399402571493429445364, 5.72968055399996647444302513531, 6.53867866205134044014860601869, 7.30961884421384020737497138060, 7.80784281279442518556758147562
