| L(s) = 1 | + 2·2-s + 5.70·3-s + 4·4-s − 5·5-s + 11.4·6-s + 7.84·7-s + 8·8-s + 5.50·9-s − 10·10-s − 11·11-s + 22.8·12-s − 49.1·13-s + 15.6·14-s − 28.5·15-s + 16·16-s − 17·17-s + 11.0·18-s − 137.·19-s − 20·20-s + 44.7·21-s − 22·22-s + 53.8·23-s + 45.6·24-s + 25·25-s − 98.3·26-s − 122.·27-s + 31.3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.09·3-s + 0.5·4-s − 0.447·5-s + 0.775·6-s + 0.423·7-s + 0.353·8-s + 0.203·9-s − 0.316·10-s − 0.301·11-s + 0.548·12-s − 1.04·13-s + 0.299·14-s − 0.490·15-s + 0.250·16-s − 0.242·17-s + 0.144·18-s − 1.66·19-s − 0.223·20-s + 0.464·21-s − 0.213·22-s + 0.487·23-s + 0.387·24-s + 0.200·25-s − 0.741·26-s − 0.873·27-s + 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 - 5.70T + 27T^{2} \) |
| 7 | \( 1 - 7.84T + 343T^{2} \) |
| 13 | \( 1 + 49.1T + 2.19e3T^{2} \) |
| 19 | \( 1 + 137.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 53.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 240.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 96.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 242.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 93.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 422.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 653.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 429.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 863.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 787.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 898.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 388.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 718.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−8.441850188767817245920083684070, −7.76616853478208345556367499588, −6.97586316781732501395377203783, −6.11789441239740174704509970545, −4.82518050967443665318773262729, −4.46831110176123604631740671539, −3.27042890004514714045232248540, −2.64727118811246982952582643496, −1.74608659198008072340169251895, 0,
1.74608659198008072340169251895, 2.64727118811246982952582643496, 3.27042890004514714045232248540, 4.46831110176123604631740671539, 4.82518050967443665318773262729, 6.11789441239740174704509970545, 6.97586316781732501395377203783, 7.76616853478208345556367499588, 8.441850188767817245920083684070
