| L(s) = 1 | + (1.22 + 1.22i)3-s + (−1.22 + 1.22i)7-s + 2.99i·9-s + (3.67 + 3.67i)13-s − i·19-s − 2.99·21-s + (−3.67 + 3.67i)27-s − 7·31-s + (−4.89 + 4.89i)37-s + 9i·39-s + (8.57 + 8.57i)43-s + 4i·49-s + (1.22 − 1.22i)57-s − 13·61-s + (−3.67 − 3.67i)63-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.462 + 0.462i)7-s + 0.999i·9-s + (1.01 + 1.01i)13-s − 0.229i·19-s − 0.654·21-s + (−0.707 + 0.707i)27-s − 1.25·31-s + (−0.805 + 0.805i)37-s + 1.44i·39-s + (1.30 + 1.30i)43-s + 0.571i·49-s + (0.162 − 0.162i)57-s − 1.66·61-s + (−0.462 − 0.462i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.826496321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.826496321\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (1.22 - 1.22i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-3.67 - 3.67i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + (4.89 - 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-8.57 - 8.57i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + (-11.0 + 11.0i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-9.79 - 9.79i)T + 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-13.4 + 13.4i)T - 97iT^{2} \) |
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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
−9.730346021418219575226368824571, −9.197983190365234507355877167180, −8.603411851190894362790421988814, −7.69343042145633088258225948033, −6.65373616702408974849958648322, −5.76845981431194705053831199387, −4.69769972474846764067093226777, −3.81736524990983242471248363028, −2.95280983750729495426076496258, −1.77087649217755426898903470981,
0.72206556061217361207384685489, 2.05968572713021108732908885601, 3.33454330567655595976170662630, 3.86750781616940135731603473403, 5.45370120231366968070528596503, 6.25905167090188330134935932664, 7.17377193137102569124923585358, 7.79800579939007418375151943310, 8.683247290732687246747399615238, 9.293188331098242839132306752721
