| L(s) = 1 | + 6.38·3-s − 5·5-s + 5.78·7-s + 13.8·9-s + 3.04·11-s − 41.8·13-s − 31.9·15-s + 89.8·17-s − 102.·19-s + 36.9·21-s + 23·23-s + 25·25-s − 84.2·27-s − 77.9·29-s + 17.8·31-s + 19.4·33-s − 28.9·35-s + 236.·37-s − 267.·39-s − 318.·41-s + 31.9·43-s − 69.0·45-s − 239.·47-s − 309.·49-s + 573.·51-s + 498.·53-s − 15.2·55-s + ⋯ |
| L(s) = 1 | + 1.22·3-s − 0.447·5-s + 0.312·7-s + 0.511·9-s + 0.0833·11-s − 0.892·13-s − 0.549·15-s + 1.28·17-s − 1.23·19-s + 0.384·21-s + 0.208·23-s + 0.200·25-s − 0.600·27-s − 0.499·29-s + 0.103·31-s + 0.102·33-s − 0.139·35-s + 1.05·37-s − 1.09·39-s − 1.21·41-s + 0.113·43-s − 0.228·45-s − 0.743·47-s − 0.902·49-s + 1.57·51-s + 1.29·53-s − 0.0372·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 6.38T + 27T^{2} \) |
| 7 | \( 1 - 5.78T + 343T^{2} \) |
| 11 | \( 1 - 3.04T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 89.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 77.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 17.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 31.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 498.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 276.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 86.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 104.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 489.T + 9.12e5T^{2} \) |
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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
−8.416026764307473197896907477779, −7.83152215281260240429038708641, −7.26043665492299661834676436817, −6.16086377365967047675790692141, −5.07800594648821061468075518410, −4.19240540028261939876558265221, −3.31976118223654206163838888206, −2.54159540047644210319237243896, −1.52450928090702140004151685320, 0,
1.52450928090702140004151685320, 2.54159540047644210319237243896, 3.31976118223654206163838888206, 4.19240540028261939876558265221, 5.07800594648821061468075518410, 6.16086377365967047675790692141, 7.26043665492299661834676436817, 7.83152215281260240429038708641, 8.416026764307473197896907477779
