| 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 + (2.25 + 2.60i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)10-s + (3.00 + 1.93i)11-s + (−0.841 − 0.540i)12-s + (1.91 − 2.20i)13-s + (3.31 + 0.971i)14-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (0.831 + 1.82i)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 + (0.853 + 0.985i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.207 − 0.238i)10-s + (0.906 + 0.582i)11-s + (−0.242 − 0.156i)12-s + (0.529 − 0.611i)13-s + (0.884 + 0.259i)14-s + (−0.0367 − 0.255i)15-s + (−0.163 − 0.188i)16-s + (0.201 + 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.31594 - 1.15637i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31594 - 1.15637i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.56 + 1.48i)T \) |
| good | 7 | \( 1 + (-2.25 - 2.60i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.00 - 1.93i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.91 + 2.20i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.831 - 1.82i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.366 - 0.803i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.83 + 4.01i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.254 + 1.76i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.67 - 0.786i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (7.69 - 2.26i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.474 + 3.29i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + (2.06 + 2.38i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.49 - 1.72i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.0614 + 0.427i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (4.69 - 3.01i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.86 + 2.48i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.637 - 1.39i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.67 - 4.24i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-16.0 - 4.69i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.55 + 10.8i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.16 - 0.929i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.43409726987653422692519541796, −9.528808169405583310188241469544, −8.561603566379378507380686389150, −7.83644293914843660252007281047, −6.48077193529824770301361320155, −5.85755461564984963055963996009, −4.91874638938870344914478619431, −3.73270115189429646965187364113, −2.31142390547038963064955938771, −1.50922597563979966837718737618,
1.60040532577742220276924873633, 3.34406630735623624463988160319, 4.16280458758557319024309781998, 5.02786426312903081579966000066, 6.09991971537523852539974608200, 6.93818755328813422614387750083, 7.944962580328243409362935146661, 8.835875843974645822124072590376, 9.721369207526169128086196992662, 10.78569400775809048411598810773
