| L(s) = 1 | + 0.966·2-s − 3·3-s − 7.06·4-s + 17.4·5-s − 2.89·6-s + 7.22·7-s − 14.5·8-s + 9·9-s + 16.8·10-s − 11·11-s + 21.1·12-s + 10.5·13-s + 6.98·14-s − 52.2·15-s + 42.4·16-s − 26.1·17-s + 8.69·18-s − 25.8·19-s − 122.·20-s − 21.6·21-s − 10.6·22-s + 9.31·23-s + 43.6·24-s + 177.·25-s + 10.1·26-s − 27·27-s − 51.0·28-s + ⋯ |
| L(s) = 1 | + 0.341·2-s − 0.577·3-s − 0.883·4-s + 1.55·5-s − 0.197·6-s + 0.390·7-s − 0.643·8-s + 0.333·9-s + 0.531·10-s − 0.301·11-s + 0.509·12-s + 0.224·13-s + 0.133·14-s − 0.898·15-s + 0.663·16-s − 0.373·17-s + 0.113·18-s − 0.312·19-s − 1.37·20-s − 0.225·21-s − 0.102·22-s + 0.0844·23-s + 0.371·24-s + 1.42·25-s + 0.0768·26-s − 0.192·27-s − 0.344·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 0.966T + 8T^{2} \) |
| 5 | \( 1 - 17.4T + 125T^{2} \) |
| 7 | \( 1 - 7.22T + 343T^{2} \) |
| 13 | \( 1 - 10.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 26.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.31T + 1.21e4T^{2} \) |
| 29 | \( 1 + 79.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 32.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 449.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 580.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 208.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 553.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.17e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 47.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 349.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 684.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 263.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 923.T + 9.12e5T^{2} \) |
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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
−8.731243837366155920204852338187, −7.56762025774529300995796550728, −6.50066181861517957743275038150, −5.83843807683344940052086624330, −5.23612232026853372537423103047, −4.60661546538928066406655992694, −3.50263565250909674739669929310, −2.24540487862065767215054101147, −1.30713742686761478407446851417, 0,
1.30713742686761478407446851417, 2.24540487862065767215054101147, 3.50263565250909674739669929310, 4.60661546538928066406655992694, 5.23612232026853372537423103047, 5.83843807683344940052086624330, 6.50066181861517957743275038150, 7.56762025774529300995796550728, 8.731243837366155920204852338187
