| L(s) = 1 | + 2.82·5-s + (1.73 + 2i)7-s − 4.89·11-s − 6·13-s + 4.89i·17-s − 8.48i·23-s + 3.00·25-s + 3.46·31-s + (4.89 + 5.65i)35-s + 6.92i·37-s + 4.89i·41-s − 3.46·43-s − 9.79·47-s + (−1.00 + 6.92i)49-s + 9.79i·53-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + (0.654 + 0.755i)7-s − 1.47·11-s − 1.66·13-s + 1.18i·17-s − 1.76i·23-s + 0.600·25-s + 0.622·31-s + (0.828 + 0.956i)35-s + 1.13i·37-s + 0.765i·41-s − 0.528·43-s − 1.42·47-s + (−0.142 + 0.989i)49-s + 1.34i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9818387879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9818387879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 4.89iT - 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 - 9.79iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−8.514639127784360933191344697648, −8.231232374325618542647148643718, −7.32167350099841683486435233188, −6.31924187897641656051054354094, −5.81330818676838847951124734140, −4.88953737151406474075273360075, −4.67422069152674377119235309260, −2.83309069221152600399256438979, −2.45566592114161714118012202897, −1.59561145858859799287572830157,
0.24044999412707426861442947642, 1.72916565447866381577097574503, 2.41445287808050129552642103096, 3.30733693801874088062718307759, 4.73855061772416024313230292932, 5.10097565592561534963349681837, 5.69569176535822646084748099187, 6.81048018823393370484907716666, 7.59684296125265011348453102534, 7.78489815713267955792671111095
