L(s) = 1 | + (−1.79 − 3.10i)5-s + (2.56 + 0.634i)7-s + (−0.200 − 0.115i)11-s + (1.16 − 0.673i)13-s − 7.94·17-s − 3.06i·19-s + (−4.87 + 2.81i)23-s + (−3.91 + 6.78i)25-s + (2.33 + 1.34i)29-s + (−1.85 + 1.07i)31-s + (−2.63 − 9.10i)35-s − 7.27·37-s + (0.813 + 1.40i)41-s + (0.927 − 1.60i)43-s + (−0.0396 + 0.0686i)47-s + ⋯ |
L(s) = 1 | + (−0.801 − 1.38i)5-s + (0.970 + 0.239i)7-s + (−0.0604 − 0.0348i)11-s + (0.323 − 0.186i)13-s − 1.92·17-s − 0.702i·19-s + (−1.01 + 0.586i)23-s + (−0.783 + 1.35i)25-s + (0.433 + 0.250i)29-s + (−0.332 + 0.192i)31-s + (−0.445 − 1.53i)35-s − 1.19·37-s + (0.127 + 0.220i)41-s + (0.141 − 0.245i)43-s + (−0.00578 + 0.0100i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03636604145\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03636604145\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.56 - 0.634i)T \) |
good | 5 | \( 1 + (1.79 + 3.10i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.200 + 0.115i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.16 + 0.673i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 7.94T + 17T^{2} \) |
| 19 | \( 1 + 3.06iT - 19T^{2} \) |
| 23 | \( 1 + (4.87 - 2.81i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.33 - 1.34i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.85 - 1.07i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 + (-0.813 - 1.40i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.927 + 1.60i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0396 - 0.0686i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 11.6iT - 53T^{2} \) |
| 59 | \( 1 + (6.48 + 11.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.729 - 0.420i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.05 - 8.75i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.47iT - 71T^{2} \) |
| 73 | \( 1 - 7.97iT - 73T^{2} \) |
| 79 | \( 1 + (-3.30 + 5.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.41 + 11.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3.56T + 89T^{2} \) |
| 97 | \( 1 + (13.1 + 7.58i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.366995743291697575028955987324, −7.73466811533872887513984326624, −6.85992671008239566929976550087, −5.80296408907730597840850396016, −4.92024402663796434683190560058, −4.52297090109554227376793909711, −3.70231326682026382942684069120, −2.27546095256377950034543159290, −1.30137787318036536654456526203, −0.01133854323153378115119647969,
1.84233969783852432490687877820, 2.65027649954987705062798511874, 3.87123807450169305286071488573, 4.22357757623777762530787154218, 5.33226692366055192283408745260, 6.53472842924795224865873450692, 6.78126791612008552142747553840, 7.79852267875711366568991728439, 8.201340138303362424698122218005, 9.041260031637739255833426035817