| L(s) = 1 | + (−1.25 − 0.643i)2-s + (−0.424 + 0.213i)3-s + (1.17 + 1.62i)4-s + (1.46 + 3.30i)5-s + (0.672 + 0.00401i)6-s + (−3.97 − 2.38i)7-s + (−0.433 − 2.79i)8-s + (−1.65 + 2.22i)9-s + (0.281 − 5.10i)10-s + (3.80 + 2.97i)11-s + (−0.843 − 0.437i)12-s + (−2.07 + 1.97i)13-s + (3.47 + 5.55i)14-s + (−1.32 − 1.09i)15-s + (−1.25 + 3.79i)16-s + (−3.03 − 3.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.890 − 0.454i)2-s + (−0.245 + 0.123i)3-s + (0.586 + 0.810i)4-s + (0.655 + 1.47i)5-s + (0.274 + 0.00163i)6-s + (−1.50 − 0.899i)7-s + (−0.153 − 0.988i)8-s + (−0.550 + 0.742i)9-s + (0.0889 − 1.61i)10-s + (1.14 + 0.896i)11-s + (−0.243 − 0.126i)12-s + (−0.576 + 0.549i)13-s + (0.927 + 1.48i)14-s + (−0.342 − 0.281i)15-s + (−0.313 + 0.949i)16-s + (−0.735 − 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0469766 + 0.277021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0469766 + 0.277021i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 + 0.643i)T \) |
| good | 3 | \( 1 + (0.424 - 0.213i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-1.46 - 3.30i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (3.97 + 2.38i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-3.80 - 2.97i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (2.07 - 1.97i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (3.03 + 3.69i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (1.00 + 5.79i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (3.10 + 1.47i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-0.753 - 0.426i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (3.72 - 0.740i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (1.11 + 0.705i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (3.62 + 4.00i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-2.55 - 7.74i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (5.50 + 2.94i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (8.39 - 4.75i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (6.18 - 6.49i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.499 - 6.77i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-3.66 - 3.15i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-12.2 - 1.81i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (1.77 - 1.06i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (7.54 + 2.28i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (10.7 - 6.78i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-12.1 + 5.74i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-6.84 - 4.57i)T + (37.1 + 89.6i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.03226619077089895458679171751, −10.33442808325192452239920166214, −9.660732561282164551333433668372, −9.112596741392215921591829986623, −7.33989002829937858542910749586, −6.86153005064139624836687666438, −6.34952761326856436116593610416, −4.34129187803670811590553721095, −3.06190415841761582915116318770, −2.23410739889664173309078415543,
0.21240664906745241994738361041, 1.76428022092917088161884257895, 3.49758600160715753097857953589, 5.35929694085709986550588913536, 6.16239676381266104389662408179, 6.36603473225441815525328530449, 8.217999928923604446571732453947, 8.834085289029742282399917425244, 9.426766930855304922078578323500, 10.00916471434313260445244822707
