L(s) = 1 | − 3.46i·5-s − 7-s − 3.31i·11-s − 3.10i·13-s − 1.40·17-s + 4.80i·19-s − 8.79·23-s − 6.99·25-s + 9.87i·29-s + 7.83·31-s + 3.46i·35-s − 5.42i·37-s − 11.5·41-s + 7.03i·43-s − 11.2·47-s + ⋯ |
L(s) = 1 | − 1.54i·5-s − 0.377·7-s − 1.00i·11-s − 0.862i·13-s − 0.340·17-s + 1.10i·19-s − 1.83·23-s − 1.39·25-s + 1.83i·29-s + 1.40·31-s + 0.585i·35-s − 0.891i·37-s − 1.80·41-s + 1.07i·43-s − 1.64·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1284181636\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1284181636\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 3.31iT - 11T^{2} \) |
| 13 | \( 1 + 3.10iT - 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 - 4.80iT - 19T^{2} \) |
| 23 | \( 1 + 8.79T + 23T^{2} \) |
| 29 | \( 1 - 9.87iT - 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 + 5.42iT - 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 7.03iT - 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 6.51iT - 53T^{2} \) |
| 59 | \( 1 + 3.89iT - 59T^{2} \) |
| 61 | \( 1 - 9.42iT - 61T^{2} \) |
| 67 | \( 1 + 0.909iT - 67T^{2} \) |
| 71 | \( 1 - 6.42T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.370iT - 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−8.099248021732456278748026967184, −8.085653385735728000510070797645, −6.69172438619381469705982458802, −6.02073519200834462002136131662, −5.36990009634371323158104395928, −4.78042440658428065212047951347, −3.80330426584092474486455147774, −3.21615100795973700706195336373, −1.89377963050074349213719889046, −1.00077602996809769598294945907,
0.03490738073083201860118625796, 1.95364650396496352615502571638, 2.45146797888473917363368237887, 3.36770603786770231006670321462, 4.19270096931919908117392773903, 4.85004619147125529753434033495, 6.11465471235099183549130646197, 6.58370666543008753645378767009, 6.94976305775526383474054784198, 7.80509584684739649973320202987