L(s) = 1 | + (1.66 − 0.474i)3-s − 2·4-s − 4.10i·5-s − 1.56·7-s + (2.55 − 1.57i)9-s + (−3.33 + 0.948i)12-s + (−1.94 − 6.84i)15-s + 4·16-s + 7.68i·17-s + 8.43·19-s + 8.21i·20-s + (−2.60 + 0.740i)21-s − 11.8·25-s + (3.50 − 3.84i)27-s + 3.12·28-s − 6.95i·29-s + ⋯ |
L(s) = 1 | + (0.961 − 0.273i)3-s − 4-s − 1.83i·5-s − 0.590·7-s + (0.850 − 0.526i)9-s + (−0.961 + 0.273i)12-s + (−0.502 − 1.76i)15-s + 16-s + 1.86i·17-s + 1.93·19-s + 1.83i·20-s + (−0.568 + 0.161i)21-s − 2.37·25-s + (0.673 − 0.739i)27-s + 0.590·28-s − 1.29i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.976756 - 0.737570i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.976756 - 0.737570i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.66 + 0.474i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 4.10iT - 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7.68iT - 17T^{2} \) |
| 19 | \( 1 - 8.43T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6.95iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 11.0iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 5.37iT - 53T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 7.68iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.87561726059081000344376921224, −12.00769722351110063017256122665, −9.824790607351474518390977732396, −9.439352475781020408947679176767, −8.416783953346164491357024358962, −7.891056847060035876497254075009, −5.90702523545162679754096326291, −4.63005458357634827071880962273, −3.61760468280945731292185628031, −1.23454570450566170948994982900,
2.90372236907266968934588291955, 3.55441166804018318099222003724, 5.23223248435632000781719646647, 6.97986019234553109497116988072, 7.61802628858760819037488479545, 9.166644548740234108960814724897, 9.753477688576576823140838610031, 10.60432784961433455169619615127, 11.89237541795715698651765447974, 13.35275353853416767277614827116