L(s) = 1 | − 5-s − 11-s − 6·13-s − 5·17-s − 5·19-s + 23-s + 25-s − 3·29-s − 6·37-s + 4·41-s − 11·43-s + 4·47-s − 53-s + 55-s − 15·59-s + 15·61-s + 6·65-s + 10·67-s + 12·71-s − 4·73-s + 2·79-s − 15·83-s + 5·85-s − 7·89-s + 5·95-s − 5·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.66·13-s − 1.21·17-s − 1.14·19-s + 0.208·23-s + 1/5·25-s − 0.557·29-s − 0.986·37-s + 0.624·41-s − 1.67·43-s + 0.583·47-s − 0.137·53-s + 0.134·55-s − 1.95·59-s + 1.92·61-s + 0.744·65-s + 1.22·67-s + 1.42·71-s − 0.468·73-s + 0.225·79-s − 1.64·83-s + 0.542·85-s − 0.741·89-s + 0.512·95-s − 0.507·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.68112022200786, −12.34930654796403, −11.71947045973199, −11.34640423529443, −10.95670470882705, −10.35370310215283, −10.11824546587686, −9.464698447264176, −9.069719209055405, −8.618201610114490, −8.036068486262485, −7.781075297110306, −7.029610285683128, −6.819122382204345, −6.424735765103511, −5.579262524352183, −5.156671664714450, −4.770630906167715, −4.169066561797941, −3.857041344402898, −3.040669088549314, −2.545768324138650, −2.104738562227933, −1.537474741567472, −0.4574138864654198, 0,
0.4574138864654198, 1.537474741567472, 2.104738562227933, 2.545768324138650, 3.040669088549314, 3.857041344402898, 4.169066561797941, 4.770630906167715, 5.156671664714450, 5.579262524352183, 6.424735765103511, 6.819122382204345, 7.029610285683128, 7.781075297110306, 8.036068486262485, 8.618201610114490, 9.069719209055405, 9.464698447264176, 10.11824546587686, 10.35370310215283, 10.95670470882705, 11.34640423529443, 11.71947045973199, 12.34930654796403, 12.68112022200786