| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s − 32·7-s + 8·8-s + 9·9-s − 60·11-s − 12·12-s + 34·13-s − 64·14-s + 16·16-s − 42·17-s + 18·18-s − 76·19-s + 96·21-s − 120·22-s − 24·24-s + 68·26-s − 27·27-s − 128·28-s + 6·29-s − 232·31-s + 32·32-s + 180·33-s − 84·34-s + 36·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.72·7-s + 0.353·8-s + 1/3·9-s − 1.64·11-s − 0.288·12-s + 0.725·13-s − 1.22·14-s + 1/4·16-s − 0.599·17-s + 0.235·18-s − 0.917·19-s + 0.997·21-s − 1.16·22-s − 0.204·24-s + 0.512·26-s − 0.192·27-s − 0.863·28-s + 0.0384·29-s − 1.34·31-s + 0.176·32-s + 0.949·33-s − 0.423·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 232 T + p^{3} T^{2} \) |
| 37 | \( 1 + 134 T + p^{3} T^{2} \) |
| 41 | \( 1 - 234 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 360 T + p^{3} T^{2} \) |
| 53 | \( 1 + 222 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 490 T + p^{3} T^{2} \) |
| 67 | \( 1 + 812 T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 746 T + p^{3} T^{2} \) |
| 79 | \( 1 - 152 T + p^{3} T^{2} \) |
| 83 | \( 1 - 804 T + p^{3} T^{2} \) |
| 89 | \( 1 + 678 T + p^{3} T^{2} \) |
| 97 | \( 1 + 2 p T + p^{3} T^{2} \) |
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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
−12.45362652417812209259387934818, −10.92670368908630044913026457451, −10.38326508680243747313010599153, −9.050920385558184121280022826880, −7.43533144734263982703172613302, −6.34135202422083732054107664604, −5.53904237853313295410264591610, −4.00809947260197858178439479759, −2.64745294896236742507632673262, 0,
2.64745294896236742507632673262, 4.00809947260197858178439479759, 5.53904237853313295410264591610, 6.34135202422083732054107664604, 7.43533144734263982703172613302, 9.050920385558184121280022826880, 10.38326508680243747313010599153, 10.92670368908630044913026457451, 12.45362652417812209259387934818
