| L(s) = 1 | + (−0.922 − 1.07i)2-s + (0.574 + 2.25i)3-s + (−0.299 + 1.97i)4-s + (−0.512 + 0.111i)5-s + (1.88 − 2.69i)6-s + (0.421 − 2.33i)7-s + (2.39 − 1.50i)8-s + (−2.10 + 1.15i)9-s + (0.591 + 0.446i)10-s + (−1.66 − 2.58i)11-s + (−4.62 + 0.462i)12-s + (−3.68 − 2.18i)13-s + (−2.89 + 1.70i)14-s + (−0.545 − 1.08i)15-s + (−3.82 − 1.18i)16-s + (0.503 − 7.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.652 − 0.758i)2-s + (0.331 + 1.30i)3-s + (−0.149 + 0.988i)4-s + (−0.229 + 0.0498i)5-s + (0.769 − 1.09i)6-s + (0.159 − 0.882i)7-s + (0.847 − 0.531i)8-s + (−0.703 + 0.384i)9-s + (0.187 + 0.141i)10-s + (−0.501 − 0.780i)11-s + (−1.33 + 0.133i)12-s + (−1.02 − 0.606i)13-s + (−0.773 + 0.454i)14-s + (−0.140 − 0.281i)15-s + (−0.955 − 0.296i)16-s + (0.121 − 1.70i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0639 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0639 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.605909 - 0.568333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.605909 - 0.568333i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.922 + 1.07i)T \) |
| 89 | \( 1 + (-2.35 - 9.13i)T \) |
| good | 3 | \( 1 + (-0.574 - 2.25i)T + (-2.63 + 1.43i)T^{2} \) |
| 5 | \( 1 + (0.512 - 0.111i)T + (4.54 - 2.07i)T^{2} \) |
| 7 | \( 1 + (-0.421 + 2.33i)T + (-6.55 - 2.44i)T^{2} \) |
| 11 | \( 1 + (1.66 + 2.58i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (3.68 + 2.18i)T + (6.23 + 11.4i)T^{2} \) |
| 17 | \( 1 + (-0.503 + 7.03i)T + (-16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (-2.28 + 2.83i)T + (-4.03 - 18.5i)T^{2} \) |
| 23 | \( 1 + (-0.00844 - 0.0785i)T + (-22.4 + 4.88i)T^{2} \) |
| 29 | \( 1 + (-3.33 - 4.80i)T + (-10.1 + 27.1i)T^{2} \) |
| 31 | \( 1 + (-0.466 + 4.33i)T + (-30.2 - 6.58i)T^{2} \) |
| 37 | \( 1 + (4.90 + 2.03i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-8.62 + 5.11i)T + (19.6 - 35.9i)T^{2} \) |
| 43 | \( 1 + (1.80 + 1.25i)T + (15.0 + 40.2i)T^{2} \) |
| 47 | \( 1 + (3.88 + 2.90i)T + (13.2 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-6.77 - 9.04i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (9.07 + 2.31i)T + (51.7 + 28.2i)T^{2} \) |
| 61 | \( 1 + (-0.0417 + 1.16i)T + (-60.8 - 4.35i)T^{2} \) |
| 67 | \( 1 + (0.573 - 3.99i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-3.14 + 14.4i)T + (-64.5 - 29.4i)T^{2} \) |
| 73 | \( 1 + (0.575 + 1.95i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (9.43 + 5.15i)T + (42.7 + 66.4i)T^{2} \) |
| 83 | \( 1 + (-2.95 + 5.89i)T + (-49.7 - 66.4i)T^{2} \) |
| 97 | \( 1 + (6.73 + 4.32i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.25085104271113576997545473692, −9.481720240793529629957523097035, −8.900637533029028076376253694692, −7.67711032899416755876916436522, −7.25856842147307470277903693260, −5.24682716906952274409148023695, −4.48396198858304861394273327067, −3.43722344243446750449467450992, −2.72755758450967032073391732965, −0.51967131546442354174325760368,
1.58799823255387242331263021601, 2.38884854282433966082409588677, 4.44666807827341342112524291597, 5.62088752805567506485116574524, 6.45741867146786671876563856438, 7.29769062470934908058457339800, 8.047006401317919736737218496017, 8.462209058241346709946048452140, 9.658414637637280008769596409046, 10.29411771813269978316960829474
