| L(s) = 1 | + 4.74·2-s − 2.44·3-s + 14.5·4-s + 1.86·5-s − 11.5·6-s − 5.76·7-s + 31.0·8-s − 21.0·9-s + 8.86·10-s + 12.0·11-s − 35.5·12-s + 18.1·13-s − 27.3·14-s − 4.55·15-s + 31.2·16-s − 48.9·17-s − 99.9·18-s + 135.·19-s + 27.1·20-s + 14.0·21-s + 57.0·22-s − 116.·23-s − 75.9·24-s − 121.·25-s + 85.9·26-s + 117.·27-s − 83.8·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.469·3-s + 1.81·4-s + 0.166·5-s − 0.788·6-s − 0.311·7-s + 1.37·8-s − 0.779·9-s + 0.280·10-s + 0.329·11-s − 0.854·12-s + 0.386·13-s − 0.522·14-s − 0.0784·15-s + 0.488·16-s − 0.698·17-s − 1.30·18-s + 1.63·19-s + 0.303·20-s + 0.146·21-s + 0.552·22-s − 1.06·23-s − 0.645·24-s − 0.972·25-s + 0.648·26-s + 0.835·27-s − 0.566·28-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 | \( 1 - 4.74T + 8T^{2} \) |
| 3 | \( 1 + 2.44T + 27T^{2} \) |
| 5 | \( 1 - 1.86T + 125T^{2} \) |
| 7 | \( 1 + 5.76T + 343T^{2} \) |
| 11 | \( 1 - 12.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 42.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 454.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 560.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 83.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 173.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 965.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 346.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 314.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.360056470224244919543468341367, −7.35366077676600446588196074082, −6.41470068711272604345449946188, −5.95321497879242146977703987784, −5.29750755669669929941779097894, −4.42950959756517392891255352238, −3.52958888571085136745703211133, −2.81663710851876963722377680834, −1.64831088554346649268404579305, 0,
1.64831088554346649268404579305, 2.81663710851876963722377680834, 3.52958888571085136745703211133, 4.42950959756517392891255352238, 5.29750755669669929941779097894, 5.95321497879242146977703987784, 6.41470068711272604345449946188, 7.35366077676600446588196074082, 8.360056470224244919543468341367
