| L(s) = 1 | − 2-s − 3-s + 4-s − 2.27·5-s + 6-s − 8-s + 9-s + 2.27·10-s − 11-s − 12-s − 1.80·13-s + 2.27·15-s + 16-s − 5.10·17-s − 18-s + 6.90·19-s − 2.27·20-s + 22-s + 8.59·23-s + 24-s + 0.171·25-s + 1.80·26-s − 27-s − 2.82·29-s − 2.27·30-s − 3.14·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.01·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.719·10-s − 0.301·11-s − 0.288·12-s − 0.499·13-s + 0.587·15-s + 0.250·16-s − 1.23·17-s − 0.235·18-s + 1.58·19-s − 0.508·20-s + 0.213·22-s + 1.79·23-s + 0.204·24-s + 0.0343·25-s + 0.353·26-s − 0.192·27-s − 0.525·29-s − 0.415·30-s − 0.563·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2.27T + 5T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 23 | \( 1 - 8.59T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 3.14T + 31T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 - 5.10T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 - 0.0759T + 47T^{2} \) |
| 53 | \( 1 + 6.87T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 + 8.63T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 9.26T + 71T^{2} \) |
| 73 | \( 1 - 3.04T + 73T^{2} \) |
| 79 | \( 1 - 9.26T + 79T^{2} \) |
| 83 | \( 1 - 1.96T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 9.01T + 97T^{2} \) |
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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
−8.162962525073651486910198939043, −7.42324610441839151208365196043, −7.15191935613702272614193410804, −6.13379320798146501709413042451, −5.20430443150634490434156029309, −4.48559902733692913720542326892, −3.44840438050793502441983074023, −2.50362139176396811487760500654, −1.09878673122210704234196497891, 0,
1.09878673122210704234196497891, 2.50362139176396811487760500654, 3.44840438050793502441983074023, 4.48559902733692913720542326892, 5.20430443150634490434156029309, 6.13379320798146501709413042451, 7.15191935613702272614193410804, 7.42324610441839151208365196043, 8.162962525073651486910198939043
