| 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
−10.07214158430828920723683593732, −9.637237737867987003908286410191, −8.937406432807379721511668068295, −7.81758307507721995164203065664, −6.71595168720651318427060114433, −6.30111969419873959695558499225, −4.75743081925946595170405204146, −4.03329416175263954167204016299, −2.63246080980123769924725029928, −1.40051767253775882849923089256,
1.69534182797887690595467793265, 3.05515276796075634440844657230, 3.68556506626407836715348994854, 4.61733593020316879967267102870, 6.49086426920036285594949847188, 6.94470916533015248601483486586, 8.144830845805705431457229666032, 8.726443631070632544078919641788, 9.452073792867935622005925072294, 10.53946569621091503188580563552
