| L(s) = 1 | + 1.79e10·3-s − 6.39e14·5-s + 1.72e18·7-s − 6.02e18·9-s − 1.05e22·11-s + 1.50e24·13-s − 1.14e25·15-s − 5.54e26·17-s + 1.99e27·19-s + 3.10e28·21-s − 7.48e28·23-s − 7.28e29·25-s − 6.00e30·27-s − 7.64e30·29-s − 1.08e31·31-s − 1.89e32·33-s − 1.10e33·35-s + 2.93e32·37-s + 2.70e34·39-s + 7.49e34·41-s − 1.23e35·43-s + 3.85e33·45-s + 5.61e35·47-s + 7.99e35·49-s − 9.94e36·51-s + 2.77e36·53-s + 6.73e36·55-s + ⋯ |
| L(s) = 1 | + 0.990·3-s − 0.599·5-s + 1.16·7-s − 0.0183·9-s − 0.429·11-s + 1.68·13-s − 0.593·15-s − 1.94·17-s + 0.640·19-s + 1.15·21-s − 0.395·23-s − 0.640·25-s − 1.00·27-s − 0.276·29-s − 0.0932·31-s − 0.425·33-s − 0.700·35-s + 0.0563·37-s + 1.67·39-s + 1.58·41-s − 0.935·43-s + 0.0110·45-s + 0.630·47-s + 0.366·49-s − 1.92·51-s + 0.235·53-s + 0.257·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1.79e10T + 3.28e20T^{2} \) |
| 5 | \( 1 + 6.39e14T + 1.13e30T^{2} \) |
| 7 | \( 1 - 1.72e18T + 2.18e36T^{2} \) |
| 11 | \( 1 + 1.05e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 1.50e24T + 7.93e47T^{2} \) |
| 17 | \( 1 + 5.54e26T + 8.11e52T^{2} \) |
| 19 | \( 1 - 1.99e27T + 9.69e54T^{2} \) |
| 23 | \( 1 + 7.48e28T + 3.58e58T^{2} \) |
| 29 | \( 1 + 7.64e30T + 7.64e62T^{2} \) |
| 31 | \( 1 + 1.08e31T + 1.34e64T^{2} \) |
| 37 | \( 1 - 2.93e32T + 2.70e67T^{2} \) |
| 41 | \( 1 - 7.49e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 1.23e35T + 1.73e70T^{2} \) |
| 47 | \( 1 - 5.61e35T + 7.94e71T^{2} \) |
| 53 | \( 1 - 2.77e36T + 1.39e74T^{2} \) |
| 59 | \( 1 + 1.73e38T + 1.40e76T^{2} \) |
| 61 | \( 1 + 1.16e38T + 5.87e76T^{2} \) |
| 67 | \( 1 + 9.18e38T + 3.32e78T^{2} \) |
| 71 | \( 1 - 3.76e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 9.44e39T + 1.32e80T^{2} \) |
| 79 | \( 1 + 3.86e40T + 3.96e81T^{2} \) |
| 83 | \( 1 + 1.39e41T + 3.31e82T^{2} \) |
| 89 | \( 1 - 9.05e41T + 6.66e83T^{2} \) |
| 97 | \( 1 + 1.23e42T + 2.69e85T^{2} \) |
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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
−10.90411095859696515900347127592, −9.067143802035980803245081270462, −8.341915159894626928468467216339, −7.56834966210593859296726582525, −5.96856472454495374472248989708, −4.50118757908438172541015628582, −3.61885933117728017996191418989, −2.39485455357327008323246321416, −1.43530222990690563340429859231, 0,
1.43530222990690563340429859231, 2.39485455357327008323246321416, 3.61885933117728017996191418989, 4.50118757908438172541015628582, 5.96856472454495374472248989708, 7.56834966210593859296726582525, 8.341915159894626928468467216339, 9.067143802035980803245081270462, 10.90411095859696515900347127592
