L(s) = 1 | + (−1.03 + 0.965i)2-s + (0.133 − 1.99i)4-s − 2.73i·5-s + (1.78 + 2.19i)8-s + (2.63 + 2.82i)10-s + 5.64·11-s − 1.41·13-s + (−3.96 − 0.534i)16-s + 6.19i·17-s + 4.13i·19-s + (−5.45 − 0.366i)20-s + (−5.83 + 5.45i)22-s + 5.64·23-s − 2.46·25-s + (1.46 − 1.36i)26-s + ⋯ |
L(s) = 1 | + (−0.730 + 0.683i)2-s + (0.0669 − 0.997i)4-s − 1.22i·5-s + (0.632 + 0.774i)8-s + (0.834 + 0.892i)10-s + 1.70·11-s − 0.392·13-s + (−0.991 − 0.133i)16-s + 1.50i·17-s + 0.947i·19-s + (−1.21 − 0.0818i)20-s + (−1.24 + 1.16i)22-s + 1.17·23-s − 0.492·25-s + (0.286 − 0.267i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.292532513\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292532513\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.03 - 0.965i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 6.19iT - 17T^{2} \) |
| 19 | \( 1 - 4.13iT - 19T^{2} \) |
| 23 | \( 1 - 5.64T + 23T^{2} \) |
| 29 | \( 1 + 0.378iT - 29T^{2} \) |
| 31 | \( 1 - 7.15iT - 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 - 5.26iT - 41T^{2} \) |
| 43 | \( 1 + 5.84iT - 43T^{2} \) |
| 47 | \( 1 - 2.13T + 47T^{2} \) |
| 53 | \( 1 - 0.656iT - 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 7.98iT - 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 13.8iT - 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 5.26iT - 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−8.962312972164930986804193586848, −8.800275369951959831362223732322, −7.983253786758277592944196071244, −6.96450948451454727852621134770, −6.29505689094514045250981442218, −5.42230539401445919752675001602, −4.61196189350349385458312830306, −3.68366034886239055982321966292, −1.73707827368119040034246332322, −1.06389229011593480632931069136,
0.805491881905255210414245545849, 2.27682443552204832855126895819, 3.03661816073466834470348246859, 3.89388030153895632446224973326, 4.95169261220903042354631354674, 6.48774329476034860153253980769, 6.97427603419105827123232951336, 7.51193800375801469086996724994, 8.730049038583323389541395547436, 9.407093621369379779462739383457