| L(s) = 1 | − 0.792i·5-s − 2.71i·7-s − 3.42·11-s − 3.37·13-s + 2.52i·17-s + 2.20i·19-s + 2.15·23-s + 4.37·25-s − 0.792i·29-s − 1.70i·31-s − 2.15·35-s − 4.74·37-s + 0.147i·41-s − 6.94i·43-s − 11.5·47-s + ⋯ |
| L(s) = 1 | − 0.354i·5-s − 1.02i·7-s − 1.03·11-s − 0.935·13-s + 0.612i·17-s + 0.506i·19-s + 0.448·23-s + 0.874·25-s − 0.147i·29-s − 0.306i·31-s − 0.363·35-s − 0.780·37-s + 0.0230i·41-s − 1.05i·43-s − 1.68·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6643940475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6643940475\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.792iT - 5T^{2} \) |
| 7 | \( 1 + 2.71iT - 7T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 - 2.52iT - 17T^{2} \) |
| 19 | \( 1 - 2.20iT - 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 + 0.792iT - 29T^{2} \) |
| 31 | \( 1 + 1.70iT - 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 - 0.147iT - 41T^{2} \) |
| 43 | \( 1 + 6.94iT - 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 8.51iT - 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 - 6.94iT - 67T^{2} \) |
| 71 | \( 1 + 1.75T + 71T^{2} \) |
| 73 | \( 1 + 2.37T + 73T^{2} \) |
| 79 | \( 1 + 10.1iT - 79T^{2} \) |
| 83 | \( 1 + 7.25T + 83T^{2} \) |
| 89 | \( 1 - 5.34iT - 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.326493679874333444950038151404, −7.58195988843365823192627156650, −7.15584909282010131599064097103, −6.27175206034477310173500641090, −5.31881743124794443790341158007, −4.79204011777171957641542659141, −3.97473668173164542707511579981, −3.10478102916842886418263470872, −2.12345564708621070568404488300, −0.988203917671832877879560020735,
0.19355954126165350744167168883, 1.82788378937035060913728088179, 2.81655188302739945688851103377, 3.07300685301956163749486331804, 4.60330586697378918822713696629, 5.11166169177837329374473162232, 5.72003136170259741732624230784, 6.79044654057206631857872651635, 7.16571943070963821457660443087, 8.133483049851798907773287752145
