| L(s) = 1 | + 2-s + 3-s + 4-s + 0.866·5-s + 6-s + 8-s + 9-s + 0.866·10-s − 4.21·11-s + 12-s − 3.03·13-s + 0.866·15-s + 16-s − 2.13·17-s + 18-s − 19-s + 0.866·20-s − 4.21·22-s − 8.04·23-s + 24-s − 4.24·25-s − 3.03·26-s + 27-s + 5.24·29-s + 0.866·30-s − 8.73·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.387·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.273·10-s − 1.27·11-s + 0.288·12-s − 0.841·13-s + 0.223·15-s + 0.250·16-s − 0.517·17-s + 0.235·18-s − 0.229·19-s + 0.193·20-s − 0.898·22-s − 1.67·23-s + 0.204·24-s − 0.849·25-s − 0.595·26-s + 0.192·27-s + 0.974·29-s + 0.158·30-s − 1.56·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.866T + 5T^{2} \) |
| 11 | \( 1 + 4.21T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 23 | \( 1 + 8.04T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 + 4.90T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 - 7.52T + 43T^{2} \) |
| 47 | \( 1 + 1.66T + 47T^{2} \) |
| 53 | \( 1 + 4.03T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 - 8.64T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 1.61T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−7.75665598596549831666105277433, −7.13721849994382759383107587799, −6.23021509802015710916063418455, −5.56997123421471866482807678072, −4.84256560504422863522093536175, −4.11268745468297858220785952148, −3.22989202378369474978213193913, −2.33545594500534476337899454516, −1.89908141749717862590629656859, 0,
1.89908141749717862590629656859, 2.33545594500534476337899454516, 3.22989202378369474978213193913, 4.11268745468297858220785952148, 4.84256560504422863522093536175, 5.56997123421471866482807678072, 6.23021509802015710916063418455, 7.13721849994382759383107587799, 7.75665598596549831666105277433
