| L(s) = 1 | − 2-s − 3-s + 4-s + 3.29·5-s + 6-s − 8-s + 9-s − 3.29·10-s − 11-s − 12-s − 6.06·13-s − 3.29·15-s + 16-s + 6.11·17-s − 18-s − 0.0511·19-s + 3.29·20-s + 22-s − 6.75·23-s + 24-s + 5.82·25-s + 6.06·26-s − 27-s + 2.82·29-s + 3.29·30-s − 5.87·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.47·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.04·10-s − 0.301·11-s − 0.288·12-s − 1.68·13-s − 0.849·15-s + 0.250·16-s + 1.48·17-s − 0.235·18-s − 0.0117·19-s + 0.735·20-s + 0.213·22-s − 1.40·23-s + 0.204·24-s + 1.16·25-s + 1.19·26-s − 0.192·27-s + 0.525·29-s + 0.600·30-s − 1.05·31-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 | \( 1 - 3.29T + 5T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 + 0.0511T + 19T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 + 6.11T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 + 1.22T + 47T^{2} \) |
| 53 | \( 1 - 3.00T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 + 0.951T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.278059559907465290082275721535, −7.47423827082841620935772709381, −6.87203851321549416368384114933, −5.87477844325301664105986250875, −5.52487197537645071332289976333, −4.69769880208904966411307553228, −3.24276316508585736914668113187, −2.22264811937197552883031382220, −1.51339317206990527609371692587, 0,
1.51339317206990527609371692587, 2.22264811937197552883031382220, 3.24276316508585736914668113187, 4.69769880208904966411307553228, 5.52487197537645071332289976333, 5.87477844325301664105986250875, 6.87203851321549416368384114933, 7.47423827082841620935772709381, 8.278059559907465290082275721535
