| L(s) = 1 | − 3-s − 3·5-s + 9-s + 6·11-s − 5·13-s + 3·15-s − 17-s + 4·19-s + 4·25-s − 27-s − 9·29-s − 31-s − 6·33-s − 8·37-s + 5·39-s − 9·41-s − 8·43-s − 3·45-s + 3·47-s + 51-s + 12·53-s − 18·55-s − 4·57-s − 3·59-s − 8·61-s + 15·65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s + 1.80·11-s − 1.38·13-s + 0.774·15-s − 0.242·17-s + 0.917·19-s + 4/5·25-s − 0.192·27-s − 1.67·29-s − 0.179·31-s − 1.04·33-s − 1.31·37-s + 0.800·39-s − 1.40·41-s − 1.21·43-s − 0.447·45-s + 0.437·47-s + 0.140·51-s + 1.64·53-s − 2.42·55-s − 0.529·57-s − 0.390·59-s − 1.02·61-s + 1.86·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.66811112338660, −12.84598868697683, −12.26506872619444, −12.00860430942940, −11.74418935124052, −11.33830137274747, −10.82339554775902, −10.14690018997633, −9.733555169079908, −9.130982723767914, −8.836602194873533, −8.113682580443554, −7.554538997686440, −7.118472250633608, −6.886526719086946, −6.270044908567618, −5.470201326574266, −5.100976383329399, −4.511344574222888, −3.910912169724143, −3.609394169597031, −3.040699467187793, −1.983367254882865, −1.529494115900853, −0.6182449389773422, 0,
0.6182449389773422, 1.529494115900853, 1.983367254882865, 3.040699467187793, 3.609394169597031, 3.910912169724143, 4.511344574222888, 5.100976383329399, 5.470201326574266, 6.270044908567618, 6.886526719086946, 7.118472250633608, 7.554538997686440, 8.113682580443554, 8.836602194873533, 9.130982723767914, 9.733555169079908, 10.14690018997633, 10.82339554775902, 11.33830137274747, 11.74418935124052, 12.00860430942940, 12.26506872619444, 12.84598868697683, 13.66811112338660
