| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7733394892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7733394892\) |
| \(L(\frac{3}{2})\) |
|
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
−8.167619831774906770066250152892, −7.64771371545815533075379692659, −7.05502613484201640616417410143, −6.04393582517120151943669135008, −5.43881121242587447977986127075, −4.21968943294111164440283813028, −3.68992759727321125410143351446, −3.15984377718022982854165654482, −1.64509798484352960354076600543, −0.34812622833819661908514973242,
0.70724620021290127199506607613, 2.44848134348943119606803756226, 3.08749187990076293728478503331, 3.92556771477577247232969339572, 4.79127078397310911911045571173, 5.61881285426700066061123227307, 6.36925529738154848684326186323, 7.21338368718287562269712423089, 7.88851603479103790148607641320, 8.623215551571356548691839890068
