| L(s) = 1 | − 4·19-s + 4·31-s − 12·41-s − 49-s + 24·59-s − 20·61-s − 24·71-s − 16·79-s + 12·89-s + 12·101-s − 28·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 0.917·19-s + 0.718·31-s − 1.87·41-s − 1/7·49-s + 3.12·59-s − 2.56·61-s − 2.84·71-s − 1.80·79-s + 1.27·89-s + 1.19·101-s − 2.68·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4887771792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4887771792\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.382100079990545844759016527139, −7.79584653969953619602081406860, −7.56915523152940761542252221423, −7.05668926364497742611776494818, −6.91426317433602509382078626176, −6.32657597080790702260491555873, −6.24406772921903899852710795949, −5.74437763488870429412853053691, −5.39651702837172225646394531581, −4.82220770363466096017042042723, −4.77910875495351423081792841561, −4.09142190210313933200620933421, −3.99144249593337090452951289436, −3.30408898715098432064263981032, −3.09010947827642034964368583757, −2.40333385791447483184842600513, −2.25583481823764720880552468080, −1.35870845172254980656310165912, −1.30606241297283033813531609135, −0.17588756095700603075518696620,
0.17588756095700603075518696620, 1.30606241297283033813531609135, 1.35870845172254980656310165912, 2.25583481823764720880552468080, 2.40333385791447483184842600513, 3.09010947827642034964368583757, 3.30408898715098432064263981032, 3.99144249593337090452951289436, 4.09142190210313933200620933421, 4.77910875495351423081792841561, 4.82220770363466096017042042723, 5.39651702837172225646394531581, 5.74437763488870429412853053691, 6.24406772921903899852710795949, 6.32657597080790702260491555873, 6.91426317433602509382078626176, 7.05668926364497742611776494818, 7.56915523152940761542252221423, 7.79584653969953619602081406860, 8.382100079990545844759016527139
