| L(s) = 1 | + (0.186 + 0.982i)2-s + (−2.89 + 0.0679i)3-s + (−0.930 + 0.366i)4-s + (0.352 − 1.33i)5-s + (−0.607 − 2.83i)6-s + (1.01 + 2.26i)7-s + (−0.533 − 0.845i)8-s + (5.39 − 0.253i)9-s + (1.37 + 0.0971i)10-s + (−2.76 − 3.10i)11-s + (2.67 − 1.12i)12-s + (0.287 − 4.07i)13-s + (−2.03 + 1.42i)14-s + (−0.930 + 3.89i)15-s + (0.731 − 0.681i)16-s + (4.51 + 4.01i)17-s + ⋯ |
| L(s) = 1 | + (0.131 + 0.694i)2-s + (−1.67 + 0.0392i)3-s + (−0.465 + 0.183i)4-s + (0.157 − 0.597i)5-s + (−0.247 − 1.15i)6-s + (0.384 + 0.856i)7-s + (−0.188 − 0.299i)8-s + (1.79 − 0.0843i)9-s + (0.436 + 0.0307i)10-s + (−0.832 − 0.935i)11-s + (0.771 − 0.324i)12-s + (0.0796 − 1.13i)13-s + (−0.544 + 0.379i)14-s + (−0.240 + 1.00i)15-s + (0.182 − 0.170i)16-s + (1.09 + 0.973i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.643472 + 0.512662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643472 + 0.512662i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.186 - 0.982i)T \) |
| 269 | \( 1 + (3.91 - 15.9i)T \) |
| good | 3 | \( 1 + (2.89 - 0.0679i)T + (2.99 - 0.140i)T^{2} \) |
| 5 | \( 1 + (-0.352 + 1.33i)T + (-4.34 - 2.46i)T^{2} \) |
| 7 | \( 1 + (-1.01 - 2.26i)T + (-4.65 + 5.23i)T^{2} \) |
| 11 | \( 1 + (2.76 + 3.10i)T + (-1.28 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.287 + 4.07i)T + (-12.8 - 1.82i)T^{2} \) |
| 17 | \( 1 + (-4.51 - 4.01i)T + (1.98 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.36 - 0.905i)T + (7.37 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.169 - 7.21i)T + (-22.9 + 1.07i)T^{2} \) |
| 29 | \( 1 + (-4.50 - 7.94i)T + (-14.8 + 24.8i)T^{2} \) |
| 31 | \( 1 + (9.49 + 1.11i)T + (30.1 + 7.20i)T^{2} \) |
| 37 | \( 1 + (3.22 - 6.00i)T + (-20.4 - 30.8i)T^{2} \) |
| 41 | \( 1 + (-2.25 - 0.428i)T + (38.1 + 15.0i)T^{2} \) |
| 43 | \( 1 + (-9.34 + 5.29i)T + (22.0 - 36.8i)T^{2} \) |
| 47 | \( 1 + (-12.5 - 3.62i)T + (39.7 + 25.0i)T^{2} \) |
| 53 | \( 1 + (6.57 + 8.96i)T + (-15.9 + 50.5i)T^{2} \) |
| 59 | \( 1 + (-1.00 - 2.56i)T + (-43.1 + 40.2i)T^{2} \) |
| 61 | \( 1 + (-4.14 - 5.12i)T + (-12.7 + 59.6i)T^{2} \) |
| 67 | \( 1 + (-1.35 - 0.533i)T + (49.0 + 45.6i)T^{2} \) |
| 71 | \( 1 + (2.03 - 2.91i)T + (-24.4 - 66.6i)T^{2} \) |
| 73 | \( 1 + (-5.00 + 13.6i)T + (-55.6 - 47.2i)T^{2} \) |
| 79 | \( 1 + (-4.33 - 3.33i)T + (20.1 + 76.3i)T^{2} \) |
| 83 | \( 1 + (-0.720 - 7.65i)T + (-81.5 + 15.4i)T^{2} \) |
| 89 | \( 1 + (-0.527 - 3.18i)T + (-84.2 + 28.6i)T^{2} \) |
| 97 | \( 1 + (-4.61 + 5.71i)T + (-20.3 - 94.8i)T^{2} \) |
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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
−10.92626741333792661963587262641, −10.41067497707184099797885914283, −9.147749261117213314901354018199, −8.202007268974027237538878902686, −7.32681399955157112602132420158, −5.93077585816966600062422688497, −5.40920902521538221353241333663, −5.22564642929301767950024890015, −3.47603948128307781128832311765, −1.08139959235657882790868148349,
0.74393771154463259705902975382, 2.38342015592441755923376098216, 4.21370202929193803151271595231, 4.85222356176392932884965040297, 5.87593438799176735092848974465, 6.96786118343972421841294489646, 7.54980812644120932333891125549, 9.326922402531362678439619267234, 10.26983168894997568566815956092, 10.70970275598349266916517189725
