| L(s) = 1 | + (−0.108 − 1.41i)2-s + (−1.21 − 0.211i)3-s + (−1.97 + 0.305i)4-s + (−1.04 + 3.15i)5-s + (−0.165 + 1.73i)6-s + (0.0945 − 1.92i)7-s + (0.644 + 2.75i)8-s + (−1.38 − 0.497i)9-s + (4.55 + 1.12i)10-s + (0.764 + 0.338i)11-s + (2.46 + 0.0455i)12-s + (1.18 − 1.02i)13-s + (−2.72 + 0.0751i)14-s + (1.93 − 3.61i)15-s + (3.81 − 1.20i)16-s + (2.62 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.0765 − 0.997i)2-s + (−0.702 − 0.121i)3-s + (−0.988 + 0.152i)4-s + (−0.465 + 1.40i)5-s + (−0.0677 + 0.709i)6-s + (0.0357 − 0.727i)7-s + (0.227 + 0.973i)8-s + (−0.463 − 0.165i)9-s + (1.44 + 0.356i)10-s + (0.230 + 0.102i)11-s + (0.712 + 0.0131i)12-s + (0.328 − 0.283i)13-s + (−0.728 + 0.0200i)14-s + (0.498 − 0.933i)15-s + (0.953 − 0.301i)16-s + (0.636 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.652607 - 0.527377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.652607 - 0.527377i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.108 + 1.41i)T \) |
| good | 3 | \( 1 + (1.21 + 0.211i)T + (2.82 + 1.01i)T^{2} \) |
| 5 | \( 1 + (1.04 - 3.15i)T + (-4.01 - 2.97i)T^{2} \) |
| 7 | \( 1 + (-0.0945 + 1.92i)T + (-6.96 - 0.686i)T^{2} \) |
| 11 | \( 1 + (-0.764 - 0.338i)T + (7.38 + 8.15i)T^{2} \) |
| 13 | \( 1 + (-1.18 + 1.02i)T + (1.90 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.62 + 1.40i)T + (9.44 - 14.1i)T^{2} \) |
| 19 | \( 1 + (-7.18 + 4.07i)T + (9.76 - 16.2i)T^{2} \) |
| 23 | \( 1 + (0.446 + 1.78i)T + (-20.2 + 10.8i)T^{2} \) |
| 29 | \( 1 + (0.719 - 0.0176i)T + (28.9 - 1.42i)T^{2} \) |
| 31 | \( 1 + (-1.17 + 0.786i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-7.92 - 0.977i)T + (35.8 + 8.99i)T^{2} \) |
| 41 | \( 1 + (3.78 - 2.80i)T + (11.9 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-1.80 + 2.55i)T + (-14.4 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-0.0188 - 0.191i)T + (-46.0 + 9.16i)T^{2} \) |
| 53 | \( 1 + (5.66 + 0.139i)T + (52.9 + 2.60i)T^{2} \) |
| 59 | \( 1 + (-1.12 + 1.30i)T + (-8.65 - 58.3i)T^{2} \) |
| 61 | \( 1 + (10.3 - 2.33i)T + (55.1 - 26.0i)T^{2} \) |
| 67 | \( 1 + (-11.7 - 7.45i)T + (28.6 + 60.5i)T^{2} \) |
| 71 | \( 1 + (-8.20 - 3.88i)T + (45.0 + 54.8i)T^{2} \) |
| 73 | \( 1 + (0.685 + 13.9i)T + (-72.6 + 7.15i)T^{2} \) |
| 79 | \( 1 + (5.87 + 7.15i)T + (-15.4 + 77.4i)T^{2} \) |
| 83 | \( 1 + (-9.63 + 1.18i)T + (80.5 - 20.1i)T^{2} \) |
| 89 | \( 1 + (-1.23 + 4.95i)T + (-78.4 - 41.9i)T^{2} \) |
| 97 | \( 1 + (-1.27 + 6.39i)T + (-89.6 - 37.1i)T^{2} \) |
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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
−10.87233174906040368613610791210, −10.18409505556411136778085886379, −9.253361548204578151331265319643, −7.898044689683889803183825888318, −7.14272185472913851444107539109, −6.04227634998277294374342545836, −4.85278894903330377067845749507, −3.53127797014627942197314003168, −2.84483647955231522700251195805, −0.77562258482538515002850500089,
1.04730257536484498243932837398, 3.67729358573510106566523809528, 4.89153876989931196709816928141, 5.50353697755113271820475336808, 6.17577136790103620244518666989, 7.69298419879115591930858118923, 8.303440418083517178199518055018, 9.125092955094088018137248910010, 9.859440162753394116343138067807, 11.28155525089375842097789907897
