| L(s) = 1 | − 15·13-s − 3·19-s − 9·27-s + 39·43-s − 42·61-s − 33·67-s + 51·73-s − 12·79-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 111·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4.16·13-s − 0.688·19-s − 1.73·27-s + 5.94·43-s − 5.37·61-s − 4.03·67-s + 5.96·73-s − 1.35·79-s − 3·121-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 + 8.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.147202402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147202402\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T^{3} + p^{3} T^{6} \) |
| 19 | \( ( 1 + T + p T^{2} )^{3} \) |
| good | 5 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 17 T^{3} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{3} \) |
| 13 | \( ( 1 + 5 T + p T^{2} )^{3}( 1 - 89 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 23 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 29 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 19 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \) |
| 37 | \( ( 1 - 323 T^{3} + p^{3} T^{6} )( 1 + 323 T^{3} + p^{3} T^{6} ) \) |
| 41 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 13 T + p T^{2} )^{3}( 1 + 449 T^{3} + p^{3} T^{6} ) \) |
| 47 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 53 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 59 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 14 T + p T^{2} )^{3}( 1 - 901 T^{3} + p^{3} T^{6} ) \) |
| 67 | \( ( 1 + 11 T + p T^{2} )^{3}( 1 + 127 T^{3} + p^{3} T^{6} ) \) |
| 71 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 17 T + p T^{2} )^{3}( 1 + 919 T^{3} + p^{3} T^{6} ) \) |
| 79 | \( ( 1 + 4 T + p T^{2} )^{3}( 1 + 503 T^{3} + p^{3} T^{6} ) \) |
| 83 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{3} \) |
| 89 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 523 T^{3} + p^{3} T^{6} )( 1 + 1853 T^{3} + p^{3} T^{6} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.36557573038158783868930921228, −5.26064188261127967646138343871, −5.20466152296892011930324892386, −4.72234029062665465591159970563, −4.51792344584795377390719354016, −4.50420625318297434403407835983, −4.48782529774830743839800350978, −4.26070063880055380678030553923, −4.21076434762285777433697652880, −3.78304727597512252087467967211, −3.71301670186060922436333326105, −3.63760929399028963353355971717, −3.00563290591730091912957140212, −2.98168497967811737288215594491, −2.73467773819184245962103259465, −2.69233575885012420473105664725, −2.66223855023195415361548740522, −2.42890887451656784483822495816, −1.87916687892896191417991267575, −1.85713694555055136596073426223, −1.69017078653650306404017444731, −1.59831968442562499563125705480, −0.73171405234377629957415947522, −0.48694326412182266934276609103, −0.42785107148980184828591342299,
0.42785107148980184828591342299, 0.48694326412182266934276609103, 0.73171405234377629957415947522, 1.59831968442562499563125705480, 1.69017078653650306404017444731, 1.85713694555055136596073426223, 1.87916687892896191417991267575, 2.42890887451656784483822495816, 2.66223855023195415361548740522, 2.69233575885012420473105664725, 2.73467773819184245962103259465, 2.98168497967811737288215594491, 3.00563290591730091912957140212, 3.63760929399028963353355971717, 3.71301670186060922436333326105, 3.78304727597512252087467967211, 4.21076434762285777433697652880, 4.26070063880055380678030553923, 4.48782529774830743839800350978, 4.50420625318297434403407835983, 4.51792344584795377390719354016, 4.72234029062665465591159970563, 5.20466152296892011930324892386, 5.26064188261127967646138343871, 5.36557573038158783868930921228
Plot not available for L-functions of degree greater than 10.
