| L(s) = 1 | − 4.93·2-s + 16.3·4-s + 14.7·5-s − 7·7-s − 41.1·8-s − 72.8·10-s − 11·11-s − 32.3·13-s + 34.5·14-s + 72.4·16-s − 45.2·17-s − 146.·19-s + 241.·20-s + 54.2·22-s + 153.·23-s + 93.2·25-s + 159.·26-s − 114.·28-s + 78.6·29-s + 106.·31-s − 27.8·32-s + 223.·34-s − 103.·35-s + 94.0·37-s + 723.·38-s − 608.·40-s + 417.·41-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 2.04·4-s + 1.32·5-s − 0.377·7-s − 1.81·8-s − 2.30·10-s − 0.301·11-s − 0.689·13-s + 0.659·14-s + 1.13·16-s − 0.645·17-s − 1.76·19-s + 2.69·20-s + 0.525·22-s + 1.38·23-s + 0.746·25-s + 1.20·26-s − 0.772·28-s + 0.503·29-s + 0.614·31-s − 0.153·32-s + 1.12·34-s − 0.499·35-s + 0.418·37-s + 3.08·38-s − 2.40·40-s + 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8765209411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8765209411\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.93T + 8T^{2} \) |
| 5 | \( 1 - 14.7T + 125T^{2} \) |
| 13 | \( 1 + 32.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 78.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 106.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 417.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 60.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 253.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 647.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 559.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 602.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 343.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 224.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 102.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 730.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 43.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 8.01T + 9.12e5T^{2} \) |
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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
−9.900629192511521821974283451247, −9.243017763076846509153279089879, −8.658423506224411195911725841008, −7.60866628890741955982625660137, −6.63443061736748684489987972802, −6.10344484070771777809950231673, −4.70889538151091165132304053734, −2.66799663689542625372205408401, −2.04751301985597671675548059527, −0.68353817373044458304890663313,
0.68353817373044458304890663313, 2.04751301985597671675548059527, 2.66799663689542625372205408401, 4.70889538151091165132304053734, 6.10344484070771777809950231673, 6.63443061736748684489987972802, 7.60866628890741955982625660137, 8.658423506224411195911725841008, 9.243017763076846509153279089879, 9.900629192511521821974283451247
