| L(s) = 1 | − 1.06·2-s − 0.463·3-s − 6.86·4-s − 5.41·5-s + 0.493·6-s − 4.55·7-s + 15.8·8-s − 26.7·9-s + 5.76·10-s + 10.2·11-s + 3.18·12-s + 4.48·13-s + 4.85·14-s + 2.50·15-s + 38.0·16-s − 32.2·17-s + 28.5·18-s − 54.9·19-s + 37.1·20-s + 2.11·21-s − 10.9·22-s + 136.·23-s − 7.34·24-s − 95.7·25-s − 4.78·26-s + 24.9·27-s + 31.2·28-s + ⋯ |
| L(s) = 1 | − 0.376·2-s − 0.0892·3-s − 0.858·4-s − 0.484·5-s + 0.0335·6-s − 0.246·7-s + 0.699·8-s − 0.992·9-s + 0.182·10-s + 0.281·11-s + 0.0765·12-s + 0.0957·13-s + 0.0926·14-s + 0.0431·15-s + 0.594·16-s − 0.459·17-s + 0.373·18-s − 0.663·19-s + 0.415·20-s + 0.0219·21-s − 0.106·22-s + 1.23·23-s − 0.0624·24-s − 0.765·25-s − 0.0360·26-s + 0.177·27-s + 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.06T + 8T^{2} \) |
| 3 | \( 1 + 0.463T + 27T^{2} \) |
| 5 | \( 1 + 5.41T + 125T^{2} \) |
| 7 | \( 1 + 4.55T + 343T^{2} \) |
| 11 | \( 1 - 10.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.48T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 27.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 285.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 62.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 561.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 381.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 167.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 328.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 496.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 374.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 270.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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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
−8.498445013389561410417509343549, −7.977137751764783111317984630816, −7.00742182759398613660131717180, −6.05608872819267912402727395148, −5.21842636428464266361694224396, −4.29670276535452116362888208760, −3.56466514207483743500791784885, −2.40686046029257555583876039135, −0.906478377069589473909150284532, 0,
0.906478377069589473909150284532, 2.40686046029257555583876039135, 3.56466514207483743500791784885, 4.29670276535452116362888208760, 5.21842636428464266361694224396, 6.05608872819267912402727395148, 7.00742182759398613660131717180, 7.977137751764783111317984630816, 8.498445013389561410417509343549
