L(s) = 1 | − 6·9-s − 24·13-s + 84·25-s − 152·37-s + 20·49-s − 88·61-s − 120·73-s − 45·81-s + 360·97-s + 552·109-s + 144·117-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 316·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 2/3·9-s − 1.84·13-s + 3.35·25-s − 4.10·37-s + 0.408·49-s − 1.44·61-s − 1.64·73-s − 5/9·81-s + 3.71·97-s + 5.06·109-s + 1.23·117-s + 0.429·121-s + 0.00787·127-s + 0.00763·131-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 − 1.86·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.250694731\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250694731\) |
\(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 2 p T^{2} + p^{4} T^{4} \) |
good | 5 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 614 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 714 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 950 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3330 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 3590 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5418 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6746 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 4090 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9218 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 11510 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 8378 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15810 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.10689332168073412800586003237, −10.01452804500833491457565862427, −9.398825701952516204538508999135, −9.016843606384522937152605004977, −8.886105192576732900615833238228, −8.812947937059081909045973510623, −8.409685389483895331322001040769, −8.094474269897240955774009407728, −7.46247041668435401666140127100, −7.35533197862825724121992851566, −7.03515121430873168330430166416, −6.95910415703906066547271563213, −6.33359633684317239176575653036, −6.16888777077254843860524860576, −5.53115874669343422208814807699, −5.25710819859385270392919339650, −5.00158943917870760443133334421, −4.55049468591177440319928361268, −4.49778979338763824838428107830, −3.49732992569682746460584074973, −3.10409391563928110667489198519, −3.05925057455437964479519765788, −2.22366461339349790618826363007, −1.73105121822732431778926147941, −0.54318900059203704815187276873,
0.54318900059203704815187276873, 1.73105121822732431778926147941, 2.22366461339349790618826363007, 3.05925057455437964479519765788, 3.10409391563928110667489198519, 3.49732992569682746460584074973, 4.49778979338763824838428107830, 4.55049468591177440319928361268, 5.00158943917870760443133334421, 5.25710819859385270392919339650, 5.53115874669343422208814807699, 6.16888777077254843860524860576, 6.33359633684317239176575653036, 6.95910415703906066547271563213, 7.03515121430873168330430166416, 7.35533197862825724121992851566, 7.46247041668435401666140127100, 8.094474269897240955774009407728, 8.409685389483895331322001040769, 8.812947937059081909045973510623, 8.886105192576732900615833238228, 9.016843606384522937152605004977, 9.398825701952516204538508999135, 10.01452804500833491457565862427, 10.10689332168073412800586003237