| L(s) = 1 | + (−0.293 − 1.70i)3-s + 5-s − 1.30i·7-s + (−2.82 + 1.00i)9-s + 0.0724·11-s − 4.54i·13-s + (−0.293 − 1.70i)15-s + 5.16i·17-s + 5.55·19-s + (−2.23 + 0.383i)21-s + 9.12i·23-s + 25-s + (2.53 + 4.53i)27-s + 9.23i·29-s + 10.8i·31-s + ⋯ |
| L(s) = 1 | + (−0.169 − 0.985i)3-s + 0.447·5-s − 0.494i·7-s + (−0.942 + 0.333i)9-s + 0.0218·11-s − 1.25i·13-s + (−0.0757 − 0.440i)15-s + 1.25i·17-s + 1.27·19-s + (−0.487 + 0.0836i)21-s + 1.90i·23-s + 0.200·25-s + (0.488 + 0.872i)27-s + 1.71i·29-s + 1.94i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.619666948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.619666948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.293 + 1.70i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (0.243 + 8.18i)T \) |
| good | 7 | \( 1 + 1.30iT - 7T^{2} \) |
| 11 | \( 1 - 0.0724T + 11T^{2} \) |
| 13 | \( 1 + 4.54iT - 13T^{2} \) |
| 17 | \( 1 - 5.16iT - 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 23 | \( 1 - 9.12iT - 23T^{2} \) |
| 29 | \( 1 - 9.23iT - 29T^{2} \) |
| 31 | \( 1 - 10.8iT - 31T^{2} \) |
| 37 | \( 1 + 0.716T + 37T^{2} \) |
| 41 | \( 1 + 7.26T + 41T^{2} \) |
| 43 | \( 1 - 3.55iT - 43T^{2} \) |
| 47 | \( 1 + 5.42iT - 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + 6.20iT - 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 71 | \( 1 - 5.55iT - 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 + 8.46iT - 79T^{2} \) |
| 83 | \( 1 - 9.48iT - 83T^{2} \) |
| 89 | \( 1 + 6.51iT - 89T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−8.432155677957059869770352102928, −7.50346172541702685220131695028, −7.18830941319697076362388700510, −6.29223060579658958283456414620, −5.41954428048083851372438836085, −5.16883160805070592768042202521, −3.49853498316786015215290169076, −3.10534747795766680628857593921, −1.65914886003555445857671053322, −1.13414710013352689960250456373,
0.51922389504407445181695205441, 2.22981802008311737388301278262, 2.82457612699071177252121559613, 4.03329051023334687110757874391, 4.60582110040991048963285450102, 5.40171226055473866710914486884, 6.09262148115183917659268530328, 6.79101074302551116703097179772, 7.77170049777742195069757007875, 8.674003539556755805374327746977
