| L(s) = 1 | − 2-s + 2.44·3-s + 4-s − 1.45·5-s − 2.44·6-s − 7-s − 8-s + 2.97·9-s + 1.45·10-s − 0.0803·11-s + 2.44·12-s − 1.21·13-s + 14-s − 3.54·15-s + 16-s − 0.678·17-s − 2.97·18-s − 1.45·20-s − 2.44·21-s + 0.0803·22-s + 3.73·23-s − 2.44·24-s − 2.89·25-s + 1.21·26-s − 0.0566·27-s − 28-s − 0.545·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.41·3-s + 0.5·4-s − 0.649·5-s − 0.998·6-s − 0.377·7-s − 0.353·8-s + 0.992·9-s + 0.459·10-s − 0.0242·11-s + 0.705·12-s − 0.336·13-s + 0.267·14-s − 0.916·15-s + 0.250·16-s − 0.164·17-s − 0.701·18-s − 0.324·20-s − 0.533·21-s + 0.0171·22-s + 0.778·23-s − 0.499·24-s − 0.578·25-s + 0.237·26-s − 0.0108·27-s − 0.188·28-s − 0.101·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 11 | \( 1 + 0.0803T + 11T^{2} \) |
| 13 | \( 1 + 1.21T + 13T^{2} \) |
| 17 | \( 1 + 0.678T + 17T^{2} \) |
| 23 | \( 1 - 3.73T + 23T^{2} \) |
| 29 | \( 1 + 0.545T + 29T^{2} \) |
| 31 | \( 1 - 4.53T + 31T^{2} \) |
| 37 | \( 1 + 4.15T + 37T^{2} \) |
| 41 | \( 1 - 2.81T + 41T^{2} \) |
| 43 | \( 1 + 7.05T + 43T^{2} \) |
| 47 | \( 1 - 2.51T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 5.91T + 59T^{2} \) |
| 61 | \( 1 - 7.79T + 61T^{2} \) |
| 67 | \( 1 + 7.30T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 1.07T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 8.57T + 89T^{2} \) |
| 97 | \( 1 - 6.23T + 97T^{2} \) |
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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
−7.942443210924831958855777519375, −7.43194746678973635641516415956, −6.82307763551180737240926448673, −5.86982924753037972689694186229, −4.73133360491030915727559139991, −3.84042025801411756844707414301, −3.12693948557102798098080733685, −2.49305648688297700322915292228, −1.46345818615564069555408690538, 0,
1.46345818615564069555408690538, 2.49305648688297700322915292228, 3.12693948557102798098080733685, 3.84042025801411756844707414301, 4.73133360491030915727559139991, 5.86982924753037972689694186229, 6.82307763551180737240926448673, 7.43194746678973635641516415956, 7.942443210924831958855777519375
