| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 3·9-s − 10-s − 2·13-s − 16-s − 6·17-s + 3·18-s + 4·19-s − 20-s + 4·23-s + 25-s + 2·26-s − 6·29-s − 8·31-s − 5·32-s + 6·34-s + 3·36-s − 2·37-s − 4·38-s + 3·40-s − 2·41-s − 4·43-s − 3·45-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 9-s − 0.316·10-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s − 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 1.11·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.474·40-s − 0.312·41-s − 0.609·43-s − 0.447·45-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.02041446476298683486486191886, −9.196982768972178400591968116867, −8.758832935924856173060099057787, −7.71367077872044547263533331174, −6.79405906039010685182823940446, −5.50821445395345299948641768104, −4.77278357968932846404359363084, −3.32452113574721608186627592149, −1.86853886263346253357248352832, 0,
1.86853886263346253357248352832, 3.32452113574721608186627592149, 4.77278357968932846404359363084, 5.50821445395345299948641768104, 6.79405906039010685182823940446, 7.71367077872044547263533331174, 8.758832935924856173060099057787, 9.196982768972178400591968116867, 10.02041446476298683486486191886
