| L(s) = 1 | + (0.267 − 0.318i)2-s + (1.72 − 0.159i)3-s + (0.317 + 1.79i)4-s + (0.409 − 0.591i)6-s + (−1.29 − 0.229i)7-s + (1.37 + 0.795i)8-s + (2.94 − 0.551i)9-s + (4.90 − 1.78i)11-s + (0.834 + 3.05i)12-s + (0.0116 + 0.0138i)13-s + (−0.419 + 0.352i)14-s + (−2.81 + 1.02i)16-s + (−2.71 + 1.56i)17-s + (0.612 − 1.08i)18-s + (0.208 − 0.361i)19-s + ⋯ |
| L(s) = 1 | + (0.188 − 0.225i)2-s + (0.995 − 0.0922i)3-s + (0.158 + 0.899i)4-s + (0.167 − 0.241i)6-s + (−0.491 − 0.0866i)7-s + (0.486 + 0.281i)8-s + (0.982 − 0.183i)9-s + (1.47 − 0.537i)11-s + (0.241 + 0.881i)12-s + (0.00321 + 0.00383i)13-s + (−0.112 + 0.0941i)14-s + (−0.703 + 0.256i)16-s + (−0.658 + 0.379i)17-s + (0.144 − 0.255i)18-s + (0.0478 − 0.0829i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.45667 + 0.344915i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.45667 + 0.344915i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 + 0.159i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.267 + 0.318i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (1.29 + 0.229i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-4.90 + 1.78i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.0116 - 0.0138i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.71 - 1.56i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.208 + 0.361i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.01 - 0.179i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.98 - 5.01i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.647 - 3.67i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.83 + 2.21i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.81 - 2.36i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.84 + 7.80i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (6.99 + 1.23i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 1.30iT - 53T^{2} \) |
| 59 | \( 1 + (3.47 + 1.26i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.20 + 6.80i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (7.08 + 8.44i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.04 - 5.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.473 + 0.273i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.374 + 0.314i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.96 + 3.53i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (1.68 - 2.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.40 + 9.34i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.53946569621091503188580563552, −9.452073792867935622005925072294, −8.726443631070632544078919641788, −8.144830845805705431457229666032, −6.94470916533015248601483486586, −6.49086426920036285594949847188, −4.61733593020316879967267102870, −3.68556506626407836715348994854, −3.05515276796075634440844657230, −1.69534182797887690595467793265,
1.40051767253775882849923089256, 2.63246080980123769924725029928, 4.03329416175263954167204016299, 4.75743081925946595170405204146, 6.30111969419873959695558499225, 6.71595168720651318427060114433, 7.81758307507721995164203065664, 8.937406432807379721511668068295, 9.637237737867987003908286410191, 10.07214158430828920723683593732
