| L(s) = 1 | + 7.89i·3-s + 22.0i·5-s + 18.7·9-s − 44.7·11-s − 311. i·13-s − 173.·15-s − 71.5i·17-s + 278. i·19-s + 514.·23-s + 140.·25-s + 786. i·27-s + 866.·29-s + 683. i·31-s − 353. i·33-s + 1.94e3·37-s + ⋯ |
| L(s) = 1 | + 0.876i·3-s + 0.880i·5-s + 0.231·9-s − 0.369·11-s − 1.84i·13-s − 0.771·15-s − 0.247i·17-s + 0.772i·19-s + 0.973·23-s + 0.224·25-s + 1.07i·27-s + 1.02·29-s + 0.711i·31-s − 0.324i·33-s + 1.42·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.234674222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.234674222\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 7.89iT - 81T^{2} \) |
| 5 | \( 1 - 22.0iT - 625T^{2} \) |
| 11 | \( 1 + 44.7T + 1.46e4T^{2} \) |
| 13 | \( 1 + 311. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 71.5iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 278. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 514.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 866.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 683. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.94e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 102. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 885.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.82e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.32e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.84e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 392. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.24e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.18e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 0.214iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 1.57e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 9.05e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 3.48e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 7.94e3iT - 8.85e7T^{2} \) |
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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
−10.16481535619573091562097538517, −9.329182150138589240989953913766, −8.169991763532291768049163322270, −7.45623697460975472001908536111, −6.42188170681225951279415546133, −5.40361315980699925435865068362, −4.57370238967065105864245594185, −3.32487771454379237631578678342, −2.78659881700946745828447608017, −0.973615220221346003970235898547,
0.64299416400213664232877685176, 1.54576405224027540511037363864, 2.57823221717261895466470652007, 4.24749933732816485358260881734, 4.83690142923483645307858318477, 6.16108859568527399666622025263, 6.92002238157762017223606727348, 7.66018526343666274824301026614, 8.734809554777806890549430312919, 9.215972464744419131248741041821
