| L(s) = 1 | + (−1.72 − 0.146i)3-s − 5-s − 1.45i·7-s + (2.95 + 0.505i)9-s + 1.01i·11-s + 3.74i·13-s + (1.72 + 0.146i)15-s − 4.75i·17-s − 2.16·19-s + (−0.212 + 2.50i)21-s − 3.30·23-s + 25-s + (−5.02 − 1.30i)27-s + 3.59·29-s − 2.75i·31-s + ⋯ |
| L(s) = 1 | + (−0.996 − 0.0845i)3-s − 0.447·5-s − 0.548i·7-s + (0.985 + 0.168i)9-s + 0.304i·11-s + 1.03i·13-s + (0.445 + 0.0378i)15-s − 1.15i·17-s − 0.497·19-s + (−0.0463 + 0.546i)21-s − 0.688·23-s + 0.200·25-s + (−0.967 − 0.251i)27-s + 0.667·29-s − 0.494i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8768793371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8768793371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.146i)T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 1.45iT - 7T^{2} \) |
| 11 | \( 1 - 1.01iT - 11T^{2} \) |
| 13 | \( 1 - 3.74iT - 13T^{2} \) |
| 17 | \( 1 + 4.75iT - 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 31 | \( 1 + 2.75iT - 31T^{2} \) |
| 37 | \( 1 - 6.06iT - 37T^{2} \) |
| 41 | \( 1 - 7.32iT - 41T^{2} \) |
| 43 | \( 1 + 0.0374T + 43T^{2} \) |
| 47 | \( 1 + 5.62T + 47T^{2} \) |
| 53 | \( 1 + 4.31T + 53T^{2} \) |
| 59 | \( 1 + 11.9iT - 59T^{2} \) |
| 61 | \( 1 - 5.30iT - 61T^{2} \) |
| 67 | \( 1 - 8.03T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 11.3iT - 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 6.31iT - 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−9.625120471078868112492178040323, −8.361421122958981301545865184201, −7.59275626654194638851320188715, −6.75597741116184652687847477353, −6.37161980451479979261936717869, −5.06197977360274495132442149059, −4.53868363937306032404110546261, −3.68570043232716847730211354052, −2.19149991277073377018551821906, −0.870462147875926255756508244364,
0.50350524572257176345826306412, 1.97132099650509749959199006976, 3.37866997246513869966991488404, 4.22579450370266115154599743584, 5.22136573083537762454762280792, 5.89240149227160005637571689414, 6.54365499924677739877191881245, 7.57479355186190687122351555826, 8.289183050619146895635689944702, 9.059755785397804781880024615649
