| L(s) = 1 | − 3.93i·5-s − 1.11i·7-s + 3.26·11-s + 0.249·13-s + 5.86i·17-s + 2.19i·19-s − 5.58·23-s − 10.4·25-s + 2.72i·29-s − 10.3i·31-s − 4.36·35-s + 0.333·37-s − 6.10i·41-s − 9.82i·43-s − 9.41·47-s + ⋯ |
| L(s) = 1 | − 1.76i·5-s − 0.419i·7-s + 0.984·11-s + 0.0692·13-s + 1.42i·17-s + 0.504i·19-s − 1.16·23-s − 2.09·25-s + 0.505i·29-s − 1.86i·31-s − 0.738·35-s + 0.0547·37-s − 0.952i·41-s − 1.49i·43-s − 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.044237320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.044237320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.93iT - 5T^{2} \) |
| 7 | \( 1 + 1.11iT - 7T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 - 0.249T + 13T^{2} \) |
| 17 | \( 1 - 5.86iT - 17T^{2} \) |
| 19 | \( 1 - 2.19iT - 19T^{2} \) |
| 23 | \( 1 + 5.58T + 23T^{2} \) |
| 29 | \( 1 - 2.72iT - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 0.333T + 37T^{2} \) |
| 41 | \( 1 + 6.10iT - 41T^{2} \) |
| 43 | \( 1 + 9.82iT - 43T^{2} \) |
| 47 | \( 1 + 9.41T + 47T^{2} \) |
| 53 | \( 1 + 4.75iT - 53T^{2} \) |
| 59 | \( 1 + 6.53T + 59T^{2} \) |
| 61 | \( 1 - 2.14T + 61T^{2} \) |
| 67 | \( 1 + 0.578iT - 67T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 8.85iT - 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 + 4.75iT - 89T^{2} \) |
| 97 | \( 1 + 1.83T + 97T^{2} \) |
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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
−8.041854141922963616785820815654, −7.27693021915084764373659346934, −6.13351670788773497490271938595, −5.79602039466103377302005004701, −4.82923039082816149769250082623, −3.95392348770534855868318405328, −3.83366015400049550332734956581, −1.97825844614850684038563189977, −1.39798814681298634599253099038, −0.27104444024404468542491566356,
1.49447732025359564182199703379, 2.68669232808485576228080907723, 3.03505873011215868643238159786, 4.00535160025630541550239067314, 4.89091930491906067979457114140, 5.95396063906805296817710172930, 6.49189695675377800066030254100, 7.02999545199418909609127195146, 7.67714340818396873855441540445, 8.538263785264699936829866690786
