L(s) = 1 | − 0.617·2-s − 0.0232·3-s − 1.61·4-s + 0.0143·6-s + 4.67·7-s + 2.23·8-s − 2.99·9-s + 2.37·11-s + 0.0376·12-s + 1.52·13-s − 2.88·14-s + 1.85·16-s − 5.63·17-s + 1.85·18-s − 0.108·21-s − 1.46·22-s − 3.24·23-s − 0.0519·24-s − 0.940·26-s + 0.139·27-s − 7.56·28-s − 9.43·29-s + 4.54·31-s − 5.61·32-s − 0.0552·33-s + 3.48·34-s + 4.85·36-s + ⋯ |
L(s) = 1 | − 0.436·2-s − 0.0134·3-s − 0.809·4-s + 0.00586·6-s + 1.76·7-s + 0.790·8-s − 0.999·9-s + 0.717·11-s + 0.0108·12-s + 0.422·13-s − 0.771·14-s + 0.464·16-s − 1.36·17-s + 0.436·18-s − 0.0236·21-s − 0.313·22-s − 0.676·23-s − 0.0106·24-s − 0.184·26-s + 0.0268·27-s − 1.42·28-s − 1.75·29-s + 0.816·31-s − 0.992·32-s − 0.00962·33-s + 0.597·34-s + 0.809·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.364092614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.364092614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.617T + 2T^{2} \) |
| 3 | \( 1 + 0.0232T + 3T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 - 1.52T + 13T^{2} \) |
| 17 | \( 1 + 5.63T + 17T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 9.43T + 29T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 + 9.36T + 37T^{2} \) |
| 41 | \( 1 - 7.91T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 - 8.49T + 53T^{2} \) |
| 59 | \( 1 - 5.97T + 59T^{2} \) |
| 61 | \( 1 + 7.79T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 5.10T + 71T^{2} \) |
| 73 | \( 1 - 2.25T + 73T^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 - 1.72T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.924340643321863995703010926097, −7.31078548991605808556781702789, −6.32630621294933246757660313493, −5.50679172814301806953304857313, −5.01863980067503441943433409309, −4.17389319352353078779908119931, −3.77577568338027406725121384707, −2.32396645388729590630532065387, −1.66795130011133525179138372187, −0.62789771727100664202949126703,
0.62789771727100664202949126703, 1.66795130011133525179138372187, 2.32396645388729590630532065387, 3.77577568338027406725121384707, 4.17389319352353078779908119931, 5.01863980067503441943433409309, 5.50679172814301806953304857313, 6.32630621294933246757660313493, 7.31078548991605808556781702789, 7.924340643321863995703010926097