| L(s) = 1 | + (−0.619 − 2.93i)3-s + (4.48 − 2.21i)5-s + 4.63i·7-s + (−8.23 + 3.63i)9-s + 15.0·11-s + 3.35·13-s + (−9.27 − 11.7i)15-s − 11.1·17-s + 22.7i·19-s + (13.5 − 2.87i)21-s + 36.1·23-s + (15.2 − 19.8i)25-s + (15.7 + 21.9i)27-s + 21.5·29-s + 28.3·31-s + ⋯ |
| L(s) = 1 | + (−0.206 − 0.978i)3-s + (0.896 − 0.442i)5-s + 0.661i·7-s + (−0.914 + 0.404i)9-s + 1.37·11-s + 0.258·13-s + (−0.618 − 0.785i)15-s − 0.655·17-s + 1.19i·19-s + (0.647 − 0.136i)21-s + 1.57·23-s + (0.608 − 0.793i)25-s + (0.584 + 0.811i)27-s + 0.743·29-s + 0.915·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.291493312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.291493312\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.619 + 2.93i)T \) |
| 5 | \( 1 + (-4.48 + 2.21i)T \) |
| good | 7 | \( 1 - 4.63iT - 49T^{2} \) |
| 11 | \( 1 - 15.0T + 121T^{2} \) |
| 13 | \( 1 - 3.35T + 169T^{2} \) |
| 17 | \( 1 + 11.1T + 289T^{2} \) |
| 19 | \( 1 - 22.7iT - 361T^{2} \) |
| 23 | \( 1 - 36.1T + 529T^{2} \) |
| 29 | \( 1 - 21.5T + 841T^{2} \) |
| 31 | \( 1 - 28.3T + 961T^{2} \) |
| 37 | \( 1 + 69.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 36.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 29.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 8.07T + 3.48e3T^{2} \) |
| 61 | \( 1 - 10.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 91.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 79.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 86.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 10.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 139. iT - 9.40e3T^{2} \) |
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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
−9.549116629231616622534189223696, −8.808408468696766406803329490903, −8.290835442429729771578566736210, −6.91562867747800727992187019537, −6.36902637627506710439831905215, −5.63023575716014507201353920177, −4.64458818481942245968568619412, −3.09448502906418288521016243653, −1.89488350178151125430473261518, −1.08111210566257944593204204869,
0.979647273974667453530634249454, 2.61740551310830111786795511131, 3.68067506885081632692277042388, 4.59976814588142677498318091255, 5.49486403391489820337484973347, 6.65630184805530974446337648300, 6.94174784294414271687327005393, 8.759313871244804275008473108954, 9.074015141591102899761446149194, 9.955770060439589377603689813262
