L(s) = 1 | + 1.83·3-s − 3.37·5-s − 3.93·7-s + 0.354·9-s + 2.11·11-s − 5.61·13-s − 6.17·15-s − 5.47·17-s + 1.80·19-s − 7.20·21-s − 7.43·23-s + 6.36·25-s − 4.84·27-s − 3.93·29-s + 0.478·31-s + 3.86·33-s + 13.2·35-s + 9.53·37-s − 10.2·39-s − 11.7·41-s + 3.65·43-s − 1.19·45-s − 6.91·47-s + 8.46·49-s − 10.0·51-s + 9.78·53-s − 7.11·55-s + ⋯ |
L(s) = 1 | + 1.05·3-s − 1.50·5-s − 1.48·7-s + 0.118·9-s + 0.636·11-s − 1.55·13-s − 1.59·15-s − 1.32·17-s + 0.415·19-s − 1.57·21-s − 1.55·23-s + 1.27·25-s − 0.932·27-s − 0.730·29-s + 0.0859·31-s + 0.673·33-s + 2.24·35-s + 1.56·37-s − 1.64·39-s − 1.82·41-s + 0.557·43-s − 0.178·45-s − 1.00·47-s + 1.20·49-s − 1.40·51-s + 1.34·53-s − 0.959·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4349985612\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4349985612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - 1.83T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 - 1.80T + 19T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 - 0.478T + 31T^{2} \) |
| 37 | \( 1 - 9.53T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 + 6.91T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 7.56T + 67T^{2} \) |
| 71 | \( 1 - 0.790T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 - 2.16T + 83T^{2} \) |
| 89 | \( 1 - 5.51T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.84926796103818237096065662595, −7.24085588426285160369904622836, −6.71185396103364034581000977117, −5.89876640004530459628724386184, −4.72493162449780626016819962412, −3.99347585532199329232141942252, −3.55574173993300205235965255555, −2.80270777741492476092092323891, −2.11439223720314478376107555444, −0.28300999234073833103958439239,
0.28300999234073833103958439239, 2.11439223720314478376107555444, 2.80270777741492476092092323891, 3.55574173993300205235965255555, 3.99347585532199329232141942252, 4.72493162449780626016819962412, 5.89876640004530459628724386184, 6.71185396103364034581000977117, 7.24085588426285160369904622836, 7.84926796103818237096065662595