L(s) = 1 | − 80·19-s − 8·25-s + 160·43-s + 20·49-s − 400·67-s − 80·73-s + 160·97-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 460·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | − 4.21·19-s − 0.319·25-s + 3.72·43-s + 0.408·49-s − 5.97·67-s − 1.09·73-s + 1.64·97-s + 2.80·121-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 − 2.72·169-s + 0.00578·173-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 + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.016831232\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.016831232\) |
\(L(2)\) |
|
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 | | \( 1 \) |
good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 842 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 332 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 778 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1840 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 982 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4268 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5810 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + 100 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 4682 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 9782 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 2422 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
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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
−6.06031069412633675808882902932, −6.05946304455884958806522186364, −6.05039601933368998342131175456, −5.93155442862215245174799442768, −5.17322695514098306319998599112, −5.16293701910768687090895482061, −5.06096050539795166328913678520, −4.52036191493287287537996850941, −4.37936103391565022821156179382, −4.23580639065801223972631129085, −4.19395666683100672097590801666, −4.13727471419766797354972718425, −3.55404649740844485069620985038, −3.45674327454576437847480472955, −3.07243265542913663496465927939, −2.80799604569383465595993447916, −2.54211969231401446098746950977, −2.36341095923150852238131432571, −2.21885677719001832908850254206, −1.74839264346501482224174085955, −1.66382956466558677427684278933, −1.28980503429743713066219381100, −0.874446351895503344352321540752, −0.41339658558687446656873549677, −0.17406383151243383918410930347,
0.17406383151243383918410930347, 0.41339658558687446656873549677, 0.874446351895503344352321540752, 1.28980503429743713066219381100, 1.66382956466558677427684278933, 1.74839264346501482224174085955, 2.21885677719001832908850254206, 2.36341095923150852238131432571, 2.54211969231401446098746950977, 2.80799604569383465595993447916, 3.07243265542913663496465927939, 3.45674327454576437847480472955, 3.55404649740844485069620985038, 4.13727471419766797354972718425, 4.19395666683100672097590801666, 4.23580639065801223972631129085, 4.37936103391565022821156179382, 4.52036191493287287537996850941, 5.06096050539795166328913678520, 5.16293701910768687090895482061, 5.17322695514098306319998599112, 5.93155442862215245174799442768, 6.05039601933368998342131175456, 6.05946304455884958806522186364, 6.06031069412633675808882902932