| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 2·7-s + 9-s − 3·11-s − 2·12-s − 4·13-s + 4·14-s − 4·16-s − 8·17-s + 2·18-s − 2·21-s − 6·22-s + 23-s − 8·26-s − 27-s + 4·28-s + 29-s − 8·31-s − 8·32-s + 3·33-s − 16·34-s + 2·36-s + 7·37-s + 4·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 0.755·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 1.10·13-s + 1.06·14-s − 16-s − 1.94·17-s + 0.471·18-s − 0.436·21-s − 1.27·22-s + 0.208·23-s − 1.56·26-s − 0.192·27-s + 0.755·28-s + 0.185·29-s − 1.43·31-s − 1.41·32-s + 0.522·33-s − 2.74·34-s + 1/3·36-s + 1.15·37-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.663003477671988172128411423907, −7.57267794450033413725994868187, −6.94741208393912926694336652961, −6.03449239221672116236167058426, −5.28269572114788096995193818073, −4.68654456311297527430440074459, −4.15797257967427379453563556870, −2.79373927165665359734182776318, −2.04532514368342061778964974779, 0,
2.04532514368342061778964974779, 2.79373927165665359734182776318, 4.15797257967427379453563556870, 4.68654456311297527430440074459, 5.28269572114788096995193818073, 6.03449239221672116236167058426, 6.94741208393912926694336652961, 7.57267794450033413725994868187, 8.663003477671988172128411423907
