L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (0.415 − 0.909i)6-s + (1.26 − 1.46i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (4.28 − 2.75i)11-s + (−0.841 + 0.540i)12-s + (−1.52 − 1.75i)13-s + (−1.85 + 0.546i)14-s + (−0.142 + 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.233 − 0.512i)17-s + ⋯ |
L(s) = 1 | + (−0.594 − 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (0.429 + 0.125i)5-s + (0.169 − 0.371i)6-s + (0.479 − 0.553i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.207 − 0.238i)10-s + (1.29 − 0.830i)11-s + (−0.242 + 0.156i)12-s + (−0.422 − 0.487i)13-s + (−0.496 + 0.145i)14-s + (−0.0367 + 0.255i)15-s + (−0.163 + 0.188i)16-s + (0.0567 − 0.124i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29912 - 0.362557i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29912 - 0.362557i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-1.93 - 4.38i)T \) |
good | 7 | \( 1 + (-1.26 + 1.46i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-4.28 + 2.75i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.52 + 1.75i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.233 + 0.512i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (2.44 + 5.35i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.35 + 5.16i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.222 - 1.54i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (2.16 - 0.636i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-11.2 - 3.31i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.508 + 3.53i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 + (3.60 - 4.15i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.61 - 5.32i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.35 + 9.42i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.95 - 3.18i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-1.55 - 0.998i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.85 + 4.05i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.29 - 3.80i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.14 + 0.335i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.43 + 9.96i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (13.2 + 3.88i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.40810825598622945820491022276, −9.461690369865735222794941279514, −8.962437915440130585022079831993, −7.969940401560096393542284781608, −6.98642002261151326295613908071, −5.97627008521469595636493786602, −4.72661233978638137136748551004, −3.72362011552374322145877938818, −2.58759971087784423790224781400, −0.998347574969434158164411272927,
1.42850810231031640454503826952, 2.31246571400818632285937281788, 4.16549931291690295593336549446, 5.33440572969211468291029845512, 6.37011417915091903158497229602, 6.96027409825631173481874480486, 8.018204993004576533794805982219, 8.824861795096710508591204819815, 9.446636933136740134572833801297, 10.37531249762780637676341443860