L(s) = 1 | + (−0.415 + 0.909i)2-s + (−1.89 − 0.557i)3-s + (−0.654 − 0.755i)4-s + (1.29 − 1.49i)6-s + (0.215 − 1.49i)7-s + (0.959 − 0.281i)8-s + (2.45 + 1.57i)9-s + (0.234 + 0.512i)11-s + (0.822 + 1.80i)12-s + (−0.186 − 1.29i)13-s + (1.27 + 0.817i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (−2.45 + 1.57i)18-s + (−1.24 + 2.72i)21-s − 0.563·22-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.909i)2-s + (−1.89 − 0.557i)3-s + (−0.654 − 0.755i)4-s + (1.29 − 1.49i)6-s + (0.215 − 1.49i)7-s + (0.959 − 0.281i)8-s + (2.45 + 1.57i)9-s + (0.234 + 0.512i)11-s + (0.822 + 1.80i)12-s + (−0.186 − 1.29i)13-s + (1.27 + 0.817i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (−2.45 + 1.57i)18-s + (−1.24 + 2.72i)21-s − 0.563·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3847010893\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3847010893\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 17 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-0.755 - 0.654i)T \) |
good | 3 | \( 1 + (1.89 + 0.557i)T + (0.841 + 0.540i)T^{2} \) |
| 5 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (-0.215 + 1.49i)T + (-0.959 - 0.281i)T^{2} \) |
| 11 | \( 1 + (-0.234 - 0.512i)T + (-0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 29 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.540 + 0.158i)T + (0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (1.84 + 0.540i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
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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
−9.802090689388242653238359032824, −8.266407687813400011767628213290, −7.55738586915428014957621199319, −6.95273395663640073113133489351, −6.41973980504794377248103975320, −5.51295418548457471482317625232, −4.77602470556638337282953011794, −4.10818708773392711153971313316, −1.53215857849250564566969387977, −0.53546874341358799649209550044,
1.30322891664639714867691649907, 2.67274603875119982844365387743, 4.08650114739215400028675567149, 4.83573454634970049586349174004, 5.46002594638302295273735121350, 6.45463728708643824729109367220, 7.17497929007553338912389633076, 8.718776910261327832580107245234, 9.206981353989050905345493796349, 9.807999074909637054309132507690