| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5805285591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5805285591\) |
| \(L(\frac{3}{2})\) |
|
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
−9.422687576977095210963915821642, −8.307372599138520270846525616340, −7.50705565207251088158405577135, −7.08787575767419269011537394083, −5.89721627387279533303010670683, −5.22866315630624823397755240879, −4.33012882558652088259908790023, −2.87485216423345783022698885508, −1.66488608485280322720609565840, −0.26700128743525946645959404261,
1.71562304189961200071544134476, 3.11498564301799755441079090310, 4.36014247350366254408309390724, 4.92833393825861352073464548962, 5.85405537714350411410635522330, 6.69795574940877162864770533604, 7.63313653584872486268714050131, 8.718516703067507995452628049761, 9.450258515487579681393239321091, 10.05303052883197802613844983018
