L(s) = 1 | + 4·5-s + 5·9-s − 2·11-s + 8·19-s + 11·25-s + 2·29-s − 12·31-s − 20·41-s + 20·45-s − 49-s − 8·55-s − 12·59-s − 8·61-s − 32·71-s + 22·79-s + 16·81-s − 24·89-s + 32·95-s − 10·99-s + 30·109-s − 19·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 5/3·9-s − 0.603·11-s + 1.83·19-s + 11/5·25-s + 0.371·29-s − 2.15·31-s − 3.12·41-s + 2.98·45-s − 1/7·49-s − 1.07·55-s − 1.56·59-s − 1.02·61-s − 3.79·71-s + 2.47·79-s + 16/9·81-s − 2.54·89-s + 3.28·95-s − 1.00·99-s + 2.87·109-s − 1.72·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.278132001\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.278132001\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.01271553225125365392271835075, −11.86472219603563750029880608609, −10.73987651498033249879184501422, −10.73931340711509041282644396758, −9.973291072586787671151137302449, −9.951796029951577445884938068870, −9.282508570175823840734942219681, −9.081778348971112441835724859090, −8.263117211754458812509823761451, −7.59611645296750645328053040871, −6.97649506921161364227662129274, −6.95914517467456206402687332422, −5.92734712435679436376224320250, −5.66337032543459096211783332516, −4.92901788565123717578006711490, −4.66421356818882351463385575940, −3.49582169365356766387530324878, −2.98119450553070394676547750746, −1.77520958075780271457567887480, −1.53107027614785989238555584190,
1.53107027614785989238555584190, 1.77520958075780271457567887480, 2.98119450553070394676547750746, 3.49582169365356766387530324878, 4.66421356818882351463385575940, 4.92901788565123717578006711490, 5.66337032543459096211783332516, 5.92734712435679436376224320250, 6.95914517467456206402687332422, 6.97649506921161364227662129274, 7.59611645296750645328053040871, 8.263117211754458812509823761451, 9.081778348971112441835724859090, 9.282508570175823840734942219681, 9.951796029951577445884938068870, 9.973291072586787671151137302449, 10.73931340711509041282644396758, 10.73987651498033249879184501422, 11.86472219603563750029880608609, 12.01271553225125365392271835075