| L(s) = 1 | − 3-s − 3.16·5-s + 5.16·7-s + 9-s + 4·13-s + 3.16·15-s + 7.16·17-s − 1.16·19-s − 5.16·21-s − 23-s + 5.00·25-s − 27-s + 8.32·29-s − 6.32·31-s − 16.3·35-s − 8.32·37-s − 4·39-s − 2·41-s + 1.16·43-s − 3.16·45-s + 0.324·47-s + 19.6·49-s − 7.16·51-s + 5.48·53-s + 1.16·57-s − 8.32·59-s + 0.324·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.41·5-s + 1.95·7-s + 0.333·9-s + 1.10·13-s + 0.816·15-s + 1.73·17-s − 0.266·19-s − 1.12·21-s − 0.208·23-s + 1.00·25-s − 0.192·27-s + 1.54·29-s − 1.13·31-s − 2.75·35-s − 1.36·37-s − 0.640·39-s − 0.312·41-s + 0.177·43-s − 0.471·45-s + 0.0473·47-s + 2.80·49-s − 1.00·51-s + 0.753·53-s + 0.153·57-s − 1.08·59-s + 0.0415·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.083848795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083848795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 - 5.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 7.16T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 0.324T + 47T^{2} \) |
| 53 | \( 1 - 5.48T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 - 0.324T + 61T^{2} \) |
| 67 | \( 1 - 1.16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 + 3.67T + 97T^{2} \) |
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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
−11.86615952412879883675120220334, −11.07309711780990470135498774543, −10.44045813142318194879587015096, −8.615496669041581633535380365428, −8.028194353469524197375006004341, −7.22767067708817501952052441572, −5.64592705785796877397900285771, −4.66183592269911041341729648264, −3.65563113816889055198977811265, −1.29477599766883490247253829196,
1.29477599766883490247253829196, 3.65563113816889055198977811265, 4.66183592269911041341729648264, 5.64592705785796877397900285771, 7.22767067708817501952052441572, 8.028194353469524197375006004341, 8.615496669041581633535380365428, 10.44045813142318194879587015096, 11.07309711780990470135498774543, 11.86615952412879883675120220334
