| L(s) = 1 | + 2.18·2-s − 9.62·3-s − 3.22·4-s + 10.6·5-s − 21.0·6-s − 26.7·7-s − 24.5·8-s + 65.6·9-s + 23.3·10-s + 11·11-s + 31.0·12-s − 58.3·14-s − 102.·15-s − 27.7·16-s + 123.·17-s + 143.·18-s + 6.83·19-s − 34.4·20-s + 257.·21-s + 24.0·22-s − 164.·23-s + 236.·24-s − 10.5·25-s − 372.·27-s + 86.1·28-s − 184.·29-s − 224.·30-s + ⋯ |
| L(s) = 1 | + 0.772·2-s − 1.85·3-s − 0.403·4-s + 0.956·5-s − 1.43·6-s − 1.44·7-s − 1.08·8-s + 2.43·9-s + 0.739·10-s + 0.301·11-s + 0.746·12-s − 1.11·14-s − 1.77·15-s − 0.434·16-s + 1.76·17-s + 1.87·18-s + 0.0825·19-s − 0.385·20-s + 2.67·21-s + 0.232·22-s − 1.49·23-s + 2.00·24-s − 0.0847·25-s − 2.65·27-s + 0.581·28-s − 1.17·29-s − 1.36·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6691843742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6691843742\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.18T + 8T^{2} \) |
| 3 | \( 1 + 9.62T + 27T^{2} \) |
| 5 | \( 1 - 10.6T + 125T^{2} \) |
| 7 | \( 1 + 26.7T + 343T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.83T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 184.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 90.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 98.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 60.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 601.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 128.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 102.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 186.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 736.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 814.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 171.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 15.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 878.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 750.T + 9.12e5T^{2} \) |
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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
−9.456634985870976668680172959781, −7.87922151290882562797969250153, −6.66254397456577722476569833373, −6.22872227699418230299831382750, −5.59935812021300691151328070380, −5.21957420480317638160172957712, −4.01246405770842970339472136162, −3.33156614491702784762856759879, −1.65806484141682363241231841276, −0.37758427889913755387502725725,
0.37758427889913755387502725725, 1.65806484141682363241231841276, 3.33156614491702784762856759879, 4.01246405770842970339472136162, 5.21957420480317638160172957712, 5.59935812021300691151328070380, 6.22872227699418230299831382750, 6.66254397456577722476569833373, 7.87922151290882562797969250153, 9.456634985870976668680172959781
