| L(s) = 1 | + 2.59·2-s + 4.75·4-s + 3.02·5-s − 0.561·7-s + 7.17·8-s + 7.85·10-s − 4.23·11-s − 4.87·13-s − 1.45·14-s + 9.12·16-s + 6.39·17-s + 6.29·19-s + 14.3·20-s − 10.9·22-s + 9.21·23-s + 4.13·25-s − 12.6·26-s − 2.67·28-s − 1.14·29-s − 7.52·31-s + 9.37·32-s + 16.6·34-s − 1.69·35-s + 8.74·37-s + 16.3·38-s + 21.6·40-s − 2.81·41-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 2.37·4-s + 1.35·5-s − 0.212·7-s + 2.53·8-s + 2.48·10-s − 1.27·11-s − 1.35·13-s − 0.390·14-s + 2.28·16-s + 1.55·17-s + 1.44·19-s + 3.21·20-s − 2.34·22-s + 1.92·23-s + 0.827·25-s − 2.48·26-s − 0.504·28-s − 0.213·29-s − 1.35·31-s + 1.65·32-s + 2.85·34-s − 0.286·35-s + 1.43·37-s + 2.65·38-s + 3.42·40-s − 0.440·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.078031669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.078031669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 5 | \( 1 - 3.02T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 - 6.39T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 - 9.21T + 23T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 + 1.57T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 - 4.00T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 0.473T + 61T^{2} \) |
| 67 | \( 1 - 1.80T + 67T^{2} \) |
| 71 | \( 1 - 1.76T + 71T^{2} \) |
| 73 | \( 1 - 0.611T + 73T^{2} \) |
| 79 | \( 1 + 2.49T + 79T^{2} \) |
| 83 | \( 1 + 8.29T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 7.36T + 97T^{2} \) |
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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
−7.70491018132782515590904369770, −7.37029020929531015319271825760, −6.58313487504341549025322477364, −5.61733831243511093680881818938, −5.23645030650331559188085070094, −5.05477410837644568597202847035, −3.67367284779508150290956472665, −2.81283711061327307112697561501, −2.51001866106751498643590602679, −1.31161423143397273915926894654,
1.31161423143397273915926894654, 2.51001866106751498643590602679, 2.81283711061327307112697561501, 3.67367284779508150290956472665, 5.05477410837644568597202847035, 5.23645030650331559188085070094, 5.61733831243511093680881818938, 6.58313487504341549025322477364, 7.37029020929531015319271825760, 7.70491018132782515590904369770
