| L(s) = 1 | + (−0.714 + 1.48i)2-s + (0.502 + 1.65i)3-s + (−0.445 − 0.559i)4-s + (3.74 + 1.15i)5-s + (−2.82 − 0.438i)6-s + (0.974 − 2.45i)7-s + (−2.06 + 0.471i)8-s + (−2.49 + 1.66i)9-s + (−4.39 + 4.73i)10-s + (0.613 − 0.0459i)11-s + (0.702 − 1.02i)12-s + (−0.250 + 0.0187i)13-s + (2.95 + 3.20i)14-s + (−0.0326 + 6.79i)15-s + (1.09 − 4.79i)16-s + (2.15 + 5.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.505 + 1.04i)2-s + (0.290 + 0.956i)3-s + (−0.222 − 0.279i)4-s + (1.67 + 0.517i)5-s + (−1.15 − 0.179i)6-s + (0.368 − 0.929i)7-s + (−0.729 + 0.166i)8-s + (−0.831 + 0.555i)9-s + (−1.39 + 1.49i)10-s + (0.184 − 0.0138i)11-s + (0.202 − 0.294i)12-s + (−0.0693 + 0.00519i)13-s + (0.789 + 0.856i)14-s + (−0.00844 + 1.75i)15-s + (0.273 − 1.19i)16-s + (0.523 + 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.381038 + 1.49093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.381038 + 1.49093i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.502 - 1.65i)T \) |
| 7 | \( 1 + (-0.974 + 2.45i)T \) |
| good | 2 | \( 1 + (0.714 - 1.48i)T + (-1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (-3.74 - 1.15i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.613 + 0.0459i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (0.250 - 0.0187i)T + (12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-2.15 - 5.50i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (4.00 - 2.31i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.747 + 4.96i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-4.93 + 1.93i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + 1.59iT - 31T^{2} \) |
| 37 | \( 1 + (5.30 + 0.799i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (3.87 - 3.59i)T + (3.06 - 40.8i)T^{2} \) |
| 43 | \( 1 + (4.22 + 3.91i)T + (3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (-9.83 - 4.73i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (1.08 + 7.19i)T + (-50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-3.23 + 14.1i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (1.64 + 1.31i)T + (13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 - 3.83T + 67T^{2} \) |
| 71 | \( 1 + (-6.93 + 5.52i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.0678 - 0.00508i)T + (72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 - 3.73T + 79T^{2} \) |
| 83 | \( 1 + (0.734 - 9.79i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (0.415 + 0.283i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-3.38 - 1.95i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.95022953557765288620907830995, −10.26615671860419926791635936689, −9.781768421648977776789455263548, −8.660151150313320641453091237571, −8.043993930062360502922357806281, −6.66033387231704948012680577982, −6.11282737486115752913474125152, −5.04998442440019618302650866176, −3.63326581496741032148180706717, −2.17411488773011030797980661204,
1.22004754868921740076364165765, 2.14373511794711745122866781389, 2.88579461189099136355678459634, 5.22775628212140014644651038112, 5.95631029942958233170999334646, 6.95176047153667665866845634465, 8.551429747673223560319737954093, 9.043940892742137626718807751018, 9.711521851849556707095380802413, 10.69293579982797952583607356491
