| L(s) = 1 | − 4.08·2-s − 6.02·3-s + 8.64·4-s + 2.18·5-s + 24.5·6-s + 8.90·7-s − 2.64·8-s + 9.26·9-s − 8.90·10-s + 2.96·11-s − 52.0·12-s + 15.9·13-s − 36.3·14-s − 13.1·15-s − 58.3·16-s − 54.4·17-s − 37.8·18-s + 136.·19-s + 18.8·20-s − 53.6·21-s − 12.0·22-s + 153.·23-s + 15.9·24-s − 120.·25-s − 64.9·26-s + 106.·27-s + 77.0·28-s + ⋯ |
| L(s) = 1 | − 1.44·2-s − 1.15·3-s + 1.08·4-s + 0.195·5-s + 1.67·6-s + 0.480·7-s − 0.117·8-s + 0.343·9-s − 0.281·10-s + 0.0811·11-s − 1.25·12-s + 0.339·13-s − 0.693·14-s − 0.226·15-s − 0.912·16-s − 0.776·17-s − 0.495·18-s + 1.64·19-s + 0.211·20-s − 0.557·21-s − 0.117·22-s + 1.38·23-s + 0.135·24-s − 0.961·25-s − 0.490·26-s + 0.761·27-s + 0.519·28-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 | \( 1 + 4.08T + 8T^{2} \) |
| 3 | \( 1 + 6.02T + 27T^{2} \) |
| 5 | \( 1 - 2.18T + 125T^{2} \) |
| 7 | \( 1 - 8.90T + 343T^{2} \) |
| 11 | \( 1 - 2.96T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 13.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 306.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 27.1T + 6.89e4T^{2} \) |
| 47 | \( 1 + 621.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 97.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 417.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 608.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 772.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 858.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 14.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 872.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 444.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 150.T + 9.12e5T^{2} \) |
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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
−8.566833509568596184302026447328, −7.82441821111357764740646482229, −6.96164279944244895726644318681, −6.35986913986221465242143206640, −5.26586656914367033501165139797, −4.73976060396950090962956266904, −3.21676092851195588862113782117, −1.78984298010896111886870145981, −0.985117770328933536384458207715, 0,
0.985117770328933536384458207715, 1.78984298010896111886870145981, 3.21676092851195588862113782117, 4.73976060396950090962956266904, 5.26586656914367033501165139797, 6.35986913986221465242143206640, 6.96164279944244895726644318681, 7.82441821111357764740646482229, 8.566833509568596184302026447328
