| L(s) = 1 | − 2.60·2-s − 1.23·4-s − 11.0·5-s − 27.4·7-s + 24.0·8-s + 28.6·10-s + 11·11-s − 78.9·13-s + 71.4·14-s − 52.5·16-s + 66.9·17-s − 19·19-s + 13.6·20-s − 28.6·22-s − 132.·23-s − 3.66·25-s + 205.·26-s + 33.9·28-s + 47.5·29-s + 267.·31-s − 55.3·32-s − 174.·34-s + 302.·35-s + 223.·37-s + 49.4·38-s − 264.·40-s − 420.·41-s + ⋯ |
| L(s) = 1 | − 0.919·2-s − 0.154·4-s − 0.985·5-s − 1.48·7-s + 1.06·8-s + 0.905·10-s + 0.301·11-s − 1.68·13-s + 1.36·14-s − 0.821·16-s + 0.955·17-s − 0.229·19-s + 0.152·20-s − 0.277·22-s − 1.20·23-s − 0.0293·25-s + 1.54·26-s + 0.229·28-s + 0.304·29-s + 1.54·31-s − 0.305·32-s − 0.878·34-s + 1.46·35-s + 0.994·37-s + 0.210·38-s − 1.04·40-s − 1.60·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 2.60T + 8T^{2} \) |
| 5 | \( 1 + 11.0T + 125T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 13 | \( 1 + 78.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 66.9T + 4.91e3T^{2} \) |
| 23 | \( 1 + 132.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 47.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 420.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 458.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 53.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 415.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 515.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 433.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 594.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 792.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 177.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 606.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.39e3T + 9.12e5T^{2} \) |
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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
−8.354790540598804559121813561550, −7.88363121557951145090651168862, −7.10491072636438728338534955093, −6.38451213403882536478246050005, −5.14699678985337351482485318510, −4.21394952482096786748465459554, −3.45375265891846846689844964515, −2.33742214091775722288875261639, −0.73405864463188735573134389839, 0,
0.73405864463188735573134389839, 2.33742214091775722288875261639, 3.45375265891846846689844964515, 4.21394952482096786748465459554, 5.14699678985337351482485318510, 6.38451213403882536478246050005, 7.10491072636438728338534955093, 7.88363121557951145090651168862, 8.354790540598804559121813561550
