| L(s) = 1 | + 4·7-s + 26·19-s − 50·25-s + 44·43-s − 86·49-s − 148·61-s − 92·73-s − 242·121-s + 127-s + 131-s + 104·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 146·169-s + 173-s − 200·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4/7·7-s + 1.36·19-s − 2·25-s + 1.02·43-s − 1.75·49-s − 2.42·61-s − 1.26·73-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.781·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.863·169-s + 0.00578·173-s − 8/7·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.191227707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.191227707\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 26 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 122 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
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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
−9.013144266978583072329578246929, −8.334250963897739631860892092402, −8.007361810268010574227573822431, −7.60449867459815183143251332094, −7.59675905822747893054517567598, −7.00529757397510823557941474697, −6.54433168328556523433031073445, −5.96461097547096894978730221148, −5.89989611881296685569338479759, −5.34621053624017788457919020461, −4.96643410292395633095126028443, −4.45582537467749790195219789099, −4.21772121507232232250917605625, −3.45710881541466887365885444231, −3.34847740832089068041818314032, −2.62762603159696652539935684562, −2.18229001061301318026477935462, −1.44187198508490336775537557177, −1.28991481228132251366326940313, −0.25321990779208096679995880352,
0.25321990779208096679995880352, 1.28991481228132251366326940313, 1.44187198508490336775537557177, 2.18229001061301318026477935462, 2.62762603159696652539935684562, 3.34847740832089068041818314032, 3.45710881541466887365885444231, 4.21772121507232232250917605625, 4.45582537467749790195219789099, 4.96643410292395633095126028443, 5.34621053624017788457919020461, 5.89989611881296685569338479759, 5.96461097547096894978730221148, 6.54433168328556523433031073445, 7.00529757397510823557941474697, 7.59675905822747893054517567598, 7.60449867459815183143251332094, 8.007361810268010574227573822431, 8.334250963897739631860892092402, 9.013144266978583072329578246929
