| L(s) = 1 | + 5.21·5-s + 2·7-s + 4.78·11-s − 1.38i·13-s − 14.6i·17-s − 26.7i·19-s − 18.0i·23-s + 2.23·25-s − 25.0·29-s − 39.5·31-s + 10.4·35-s − 26i·37-s + 28.8i·41-s − 9.52i·43-s − 80.2i·47-s + ⋯ |
| L(s) = 1 | + 1.04·5-s + 0.285·7-s + 0.434·11-s − 0.106i·13-s − 0.863i·17-s − 1.40i·19-s − 0.784i·23-s + 0.0895·25-s − 0.862·29-s − 1.27·31-s + 0.298·35-s − 0.702i·37-s + 0.702i·41-s − 0.221i·43-s − 1.70i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.019177889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.019177889\) |
| \(L(2)\) |
|
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 | \( 1 - 5.21T + 25T^{2} \) |
| 7 | \( 1 - 2T + 49T^{2} \) |
| 11 | \( 1 - 4.78T + 121T^{2} \) |
| 13 | \( 1 + 1.38iT - 169T^{2} \) |
| 17 | \( 1 + 14.6iT - 289T^{2} \) |
| 19 | \( 1 + 26.7iT - 361T^{2} \) |
| 23 | \( 1 + 18.0iT - 529T^{2} \) |
| 29 | \( 1 + 25.0T + 841T^{2} \) |
| 31 | \( 1 + 39.5T + 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 28.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 9.52iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 80.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 9.79T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 67.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 102. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 21.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 140.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.476T + 6.24e3T^{2} \) |
| 83 | \( 1 - 31.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 13.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 69.2T + 9.40e3T^{2} \) |
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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
−8.918103019418486305152477750770, −7.77755055567267951671249388168, −6.98990446941189504476957074742, −6.30206806380430864324422699733, −5.38589605149984051484312263914, −4.80972668464115533390707438013, −3.66878592835752016189693852814, −2.54492544872500582145744488989, −1.77168604203948827887896980359, −0.44845564504105290805584737654,
1.47414696121921333947652408112, 1.93671339044234338154036868549, 3.34763121661332464043381698322, 4.13173490953884261968332310363, 5.30452797141835017942106605851, 5.88578240777387350640914137162, 6.53273231847022933761093829180, 7.61073445376935140464787394328, 8.211556206120427580359913648531, 9.293487671023365329850926065799
