| L(s) = 1 | + 2.57·2-s + 4.65·4-s + 1.53·5-s − 2.06·7-s + 6.84·8-s + 3.95·10-s + 4.40·11-s + 1.57·13-s − 5.33·14-s + 8.34·16-s + 1.87·17-s − 8.61·19-s + 7.13·20-s + 11.3·22-s − 5.69·23-s − 2.65·25-s + 4.06·26-s − 9.62·28-s − 3.91·29-s + 5.90·31-s + 7.83·32-s + 4.82·34-s − 3.17·35-s + 9.08·37-s − 22.2·38-s + 10.4·40-s + 5.30·41-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.32·4-s + 0.685·5-s − 0.782·7-s + 2.41·8-s + 1.24·10-s + 1.32·11-s + 0.437·13-s − 1.42·14-s + 2.08·16-s + 0.454·17-s − 1.97·19-s + 1.59·20-s + 2.42·22-s − 1.18·23-s − 0.530·25-s + 0.797·26-s − 1.81·28-s − 0.727·29-s + 1.06·31-s + 1.38·32-s + 0.828·34-s − 0.536·35-s + 1.49·37-s − 3.60·38-s + 1.65·40-s + 0.827·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.267601752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.267601752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 - 1.87T + 17T^{2} \) |
| 19 | \( 1 + 8.61T + 19T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 - 5.90T + 31T^{2} \) |
| 37 | \( 1 - 9.08T + 37T^{2} \) |
| 41 | \( 1 - 5.30T + 41T^{2} \) |
| 43 | \( 1 + 0.847T + 43T^{2} \) |
| 47 | \( 1 - 6.60T + 47T^{2} \) |
| 53 | \( 1 + 4.80T + 53T^{2} \) |
| 59 | \( 1 + 5.69T + 59T^{2} \) |
| 61 | \( 1 - 9.72T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 + 3.99T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 1.33T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−9.982377795055503074556304902748, −9.137816432192777956915081015079, −7.908177714142435138244704656157, −6.67635322255740485641973644841, −6.18738018962176181076105644254, −5.78444313208005100264768005182, −4.30614042605442085951524475463, −3.94564946953249249553842059698, −2.75365075274676892457091305597, −1.74645263781494647661305287410,
1.74645263781494647661305287410, 2.75365075274676892457091305597, 3.94564946953249249553842059698, 4.30614042605442085951524475463, 5.78444313208005100264768005182, 6.18738018962176181076105644254, 6.67635322255740485641973644841, 7.908177714142435138244704656157, 9.137816432192777956915081015079, 9.982377795055503074556304902748
