| L(s) = 1 | + (2.25 + 0.157i)2-s + (0.0971 + 2.78i)3-s + (3.06 + 0.430i)4-s + (1.13 − 1.92i)5-s + (−0.219 + 6.27i)6-s + (0.760 + 1.31i)7-s + (2.40 + 0.511i)8-s + (−4.73 + 0.330i)9-s + (2.86 − 4.15i)10-s + (0.226 + 2.15i)11-s + (−0.898 + 8.55i)12-s + (−2.83 − 4.20i)13-s + (1.50 + 3.08i)14-s + (5.46 + 2.97i)15-s + (−0.603 − 0.173i)16-s + (0.554 − 1.37i)17-s + ⋯ |
| L(s) = 1 | + (1.59 + 0.111i)2-s + (0.0560 + 1.60i)3-s + (1.53 + 0.215i)4-s + (0.509 − 0.860i)5-s + (−0.0894 + 2.56i)6-s + (0.287 + 0.497i)7-s + (0.851 + 0.180i)8-s + (−1.57 + 0.110i)9-s + (0.906 − 1.31i)10-s + (0.0681 + 0.648i)11-s + (−0.259 + 2.46i)12-s + (−0.787 − 1.16i)13-s + (0.402 + 0.824i)14-s + (1.41 + 0.769i)15-s + (−0.150 − 0.0432i)16-s + (0.134 − 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.82604 + 2.01742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.82604 + 2.01742i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.13 + 1.92i)T \) |
| 19 | \( 1 + (-4.31 + 0.621i)T \) |
| good | 2 | \( 1 + (-2.25 - 0.157i)T + (1.98 + 0.278i)T^{2} \) |
| 3 | \( 1 + (-0.0971 - 2.78i)T + (-2.99 + 0.209i)T^{2} \) |
| 7 | \( 1 + (-0.760 - 1.31i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.226 - 2.15i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (2.83 + 4.20i)T + (-4.86 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-0.554 + 1.37i)T + (-12.2 - 11.8i)T^{2} \) |
| 23 | \( 1 + (-0.414 - 0.400i)T + (0.802 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.984 + 2.43i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (3.46 - 3.84i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (5.25 + 3.81i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 0.310i)T + (34.7 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.727 - 4.12i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.69 - 4.19i)T + (-33.8 + 32.6i)T^{2} \) |
| 53 | \( 1 + (-12.7 - 1.79i)T + (50.9 + 14.6i)T^{2} \) |
| 59 | \( 1 + (2.22 + 8.90i)T + (-52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (-2.33 - 2.25i)T + (2.12 + 60.9i)T^{2} \) |
| 67 | \( 1 + (-12.5 - 7.83i)T + (29.3 + 60.2i)T^{2} \) |
| 71 | \( 1 + (14.1 + 7.52i)T + (39.7 + 58.8i)T^{2} \) |
| 73 | \( 1 + (-0.0951 + 0.140i)T + (-27.3 - 67.6i)T^{2} \) |
| 79 | \( 1 + (0.175 + 5.02i)T + (-78.8 + 5.51i)T^{2} \) |
| 83 | \( 1 + (11.9 - 13.3i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-4.54 + 1.30i)T + (75.4 - 47.1i)T^{2} \) |
| 97 | \( 1 + (4.16 - 2.60i)T + (42.5 - 87.1i)T^{2} \) |
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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
−11.40146528908114669382968145024, −10.25592775235267434089940542791, −9.571282786055991815053652206086, −8.739350311565980284875718007901, −7.34961522901865274002045091802, −5.70560923650041036682319782521, −5.23578438756095109630977252972, −4.69530307052080844309711159713, −3.63016291019984535019049301895, −2.52034492628193119471778636234,
1.72159701609959581901396902661, 2.69252876446683825001884751785, 3.83323751220619585220437413862, 5.34049719479713892601571245035, 6.14215242391936850607659967376, 7.02863478133049655585712091949, 7.39345625718197594231912056423, 8.883561895746974704519911890947, 10.32295661575243075790242125575, 11.49337657231920513523114182823
