L(s) = 1 | − 2·5-s − 6·13-s − 4·17-s + 5·25-s − 10·29-s − 4·37-s + 10·41-s + 7·49-s − 28·53-s + 10·61-s + 12·65-s − 12·73-s + 8·85-s − 20·89-s − 18·97-s − 2·101-s + 12·109-s − 14·113-s + 11·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 1.66·13-s − 0.970·17-s + 25-s − 1.85·29-s − 0.657·37-s + 1.56·41-s + 49-s − 3.84·53-s + 1.28·61-s + 1.48·65-s − 1.40·73-s + 0.867·85-s − 2.11·89-s − 1.82·97-s − 0.199·101-s + 1.14·109-s − 1.31·113-s + 121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.643592875549318202947190257217, −8.322147529835774295258516687193, −7.78945872239334069432400870310, −7.59770541603171538995517470939, −7.13284103448834137323053168164, −7.02116844692040028354355110798, −6.44597021406819334855555102172, −6.00013225684640725720855123249, −5.50467918575583979232735301346, −5.12197879180080150589992337435, −4.61522750701452278382929100407, −4.42553277927238074097862508984, −3.85023463839598664574010788232, −3.51015662420482243419935972866, −2.70080902117197413099128563541, −2.62637205874520343167341067342, −1.87434943159650414500715140578, −1.24908843676189811929719496116, 0, 0,
1.24908843676189811929719496116, 1.87434943159650414500715140578, 2.62637205874520343167341067342, 2.70080902117197413099128563541, 3.51015662420482243419935972866, 3.85023463839598664574010788232, 4.42553277927238074097862508984, 4.61522750701452278382929100407, 5.12197879180080150589992337435, 5.50467918575583979232735301346, 6.00013225684640725720855123249, 6.44597021406819334855555102172, 7.02116844692040028354355110798, 7.13284103448834137323053168164, 7.59770541603171538995517470939, 7.78945872239334069432400870310, 8.322147529835774295258516687193, 8.643592875549318202947190257217