| L(s) = 1 | + (1.70 + 0.306i)3-s − 2.09i·5-s + 0.613i·7-s + (2.81 + 1.04i)9-s − 13-s + (0.641 − 3.56i)15-s − 2.09i·17-s + 5.29i·19-s + (−0.188 + 1.04i)21-s + 6.81·23-s + 0.623·25-s + (4.47 + 2.64i)27-s + 6.92i·29-s + 5.29i·31-s + 1.28·35-s + ⋯ |
| L(s) = 1 | + (0.984 + 0.177i)3-s − 0.935i·5-s + 0.231i·7-s + (0.937 + 0.348i)9-s − 0.277·13-s + (0.165 − 0.920i)15-s − 0.507i·17-s + 1.21i·19-s + (−0.0410 + 0.228i)21-s + 1.42·23-s + 0.124·25-s + (0.860 + 0.509i)27-s + 1.28i·29-s + 0.950i·31-s + 0.216·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.781927299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781927299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 - 0.306i)T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.09iT - 5T^{2} \) |
| 7 | \( 1 - 0.613iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 2.09iT - 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 5.29iT - 31T^{2} \) |
| 37 | \( 1 - 5.62T + 37T^{2} \) |
| 41 | \( 1 + 11.1iT - 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 3.40T + 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 5.29iT - 67T^{2} \) |
| 71 | \( 1 - 3.40T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 6.51iT - 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 2.74iT - 89T^{2} \) |
| 97 | \( 1 + 5.24T + 97T^{2} \) |
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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
−8.825347259994130245188144934085, −8.443371795150982193829038309862, −7.39828287916102690955060514737, −6.89741292273336277557662540967, −5.45862395569118106323990384148, −5.00110024795910359070431098620, −4.00030666340600545359972195800, −3.18426366062425349574590546533, −2.13291767832180828889870817781, −1.06153172138685407513097687648,
1.08752942978276372344887317588, 2.60861464887077656175590124229, 2.84591243130649101789666990484, 4.04897167715966041661954310158, 4.74539473666343628817465800957, 6.14832262044939053036020968659, 6.74209868026450612599021580354, 7.52696316083618800131878604516, 7.995883760140660211355306384338, 9.062572236490262530283175290739
