L(s) = 1 | − 1.56·3-s + (2 + i)5-s + (2 + 1.73i)7-s − 0.561·9-s + 6.16·11-s + 3.56i·13-s + (−3.12 − 1.56i)15-s + 2.70·17-s − 7.12i·19-s + (−3.12 − 2.70i)21-s + 7.12·23-s + (3 + 4i)25-s + 5.56·27-s − 2.70i·29-s − 5.40·31-s + ⋯ |
L(s) = 1 | − 0.901·3-s + (0.894 + 0.447i)5-s + (0.755 + 0.654i)7-s − 0.187·9-s + 1.85·11-s + 0.987i·13-s + (−0.806 − 0.403i)15-s + 0.655·17-s − 1.63i·19-s + (−0.681 − 0.590i)21-s + 1.48·23-s + (0.600 + 0.800i)25-s + 1.07·27-s − 0.502i·29-s − 0.971·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.934689226\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934689226\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 - 3.56iT - 13T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 + 7.12iT - 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 2.70iT - 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 - 5.40iT - 41T^{2} \) |
| 43 | \( 1 + 12.3iT - 43T^{2} \) |
| 47 | \( 1 - 6.16iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 2.24iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 4.68iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 5.40iT - 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.980636941889024739118803208313, −8.802628564907776140167850107915, −7.15597519219426694423511217923, −6.73017184196174300889138115258, −5.98237791593997758216938124665, −5.24584471775610939376958371910, −4.55929520428625300070739521681, −3.26725933893178402795215182543, −2.11635264673453568202352333124, −1.13926520043558755051270648334,
0.998292322943178458003477946571, 1.56639228323421854043106160544, 3.25950986494873095899233271471, 4.21171824511750138789011556589, 5.28726077610695542151499969472, 5.61064865568573410928117264247, 6.50797721492532908001602878496, 7.23848167830876841810194718831, 8.325839440698071807295136875975, 8.916060091228540710888197110701