| L(s) = 1 | + (−0.978 − 0.207i)2-s + (−1.36 + 1.06i)3-s + (0.913 + 0.406i)4-s + (−1.18 + 1.89i)5-s + (1.55 − 0.761i)6-s + (1.57 − 2.72i)7-s + (−0.809 − 0.587i)8-s + (0.718 − 2.91i)9-s + (1.55 − 1.60i)10-s + (3.79 + 0.806i)11-s + (−1.68 + 0.421i)12-s + (0.172 − 0.0365i)13-s + (−2.10 + 2.34i)14-s + (−0.409 − 3.85i)15-s + (0.669 + 0.743i)16-s + (1.13 + 0.827i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.787 + 0.616i)3-s + (0.456 + 0.203i)4-s + (−0.529 + 0.848i)5-s + (0.635 − 0.310i)6-s + (0.595 − 1.03i)7-s + (−0.286 − 0.207i)8-s + (0.239 − 0.970i)9-s + (0.491 − 0.508i)10-s + (1.14 + 0.243i)11-s + (−0.484 + 0.121i)12-s + (0.0477 − 0.0101i)13-s + (−0.563 + 0.625i)14-s + (−0.105 − 0.994i)15-s + (0.167 + 0.185i)16-s + (0.276 + 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.639141 + 0.413109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.639141 + 0.413109i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (1.36 - 1.06i)T \) |
| 5 | \( 1 + (1.18 - 1.89i)T \) |
| good | 7 | \( 1 + (-1.57 + 2.72i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.79 - 0.806i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.172 + 0.0365i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.13 - 0.827i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.89 - 2.10i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.97 - 5.52i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.518 - 4.92i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.561 - 5.34i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.0665 - 0.204i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-4.46 + 0.949i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.219 + 0.380i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.315 - 3.00i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-4.37 + 3.17i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-12.3 + 2.62i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-9.91 - 2.10i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.278 + 2.64i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (12.8 - 9.35i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.64 - 8.14i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.269 - 2.56i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (14.9 - 6.64i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (4.58 + 14.1i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.43 + 13.6i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−11.33720926330571187043382655756, −10.25369403904843045876889812505, −9.886466255276947253525939050775, −8.604518791581369920022271795199, −7.40672230142677462969024147283, −6.88717224793630943179613920105, −5.69136871329876003190746901245, −4.17478788358713270074616192039, −3.51723544837717079474166679142, −1.27946018132243808753108746181,
0.792476015391063000608876718636, 2.16360221490720913654429208953, 4.29665043197709760428725298452, 5.45454560355816555186943891389, 6.21401535195741575283826433680, 7.38922573014951632000088706475, 8.251529292437488421792979456929, 8.909587658440842515762381104053, 9.925358931763382293252867003601, 11.27119095635141836090979541357
