| L(s) = 1 | − 36·7-s + 252·13-s + 79·16-s + 784·31-s + 828·37-s − 576·43-s + 648·49-s − 1.28e3·61-s + 1.51e3·67-s + 1.51e3·73-s − 9.07e3·91-s − 1.00e3·97-s + 3.78e3·103-s − 2.84e3·112-s + 2.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.17e4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.94·7-s + 5.37·13-s + 1.23·16-s + 4.54·31-s + 3.67·37-s − 2.04·43-s + 1.88·49-s − 2.70·61-s + 2.75·67-s + 2.42·73-s − 10.4·91-s − 1.05·97-s + 3.61·103-s − 2.39·112-s + 1.80·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 14.4·169-s + 0.000439·173-s + 0.000417·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.277669240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.277669240\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 79 T^{4} + p^{12} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 1204 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 126 T + 7938 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 16720706 T^{4} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 8818 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 102026878 T^{4} + p^{12} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3710 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 196 T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 414 T + 85698 T^{2} - 414 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 66400 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 288 T + 41472 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 19408253758 T^{4} + p^{12} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 718 p^{4} T^{4} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 339316 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 322 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 756 T + 285768 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 10150 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 756 T + 285768 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 747934 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 651490514638 T^{4} + p^{12} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 1338496 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 504 T + 127008 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−8.545552111989571715238918445272, −8.268849704843283699012750471437, −8.001510769350798589906393532469, −7.78457504866236935251638874729, −7.63988272991155122013862636542, −6.83050332634696054167295389856, −6.58995680535944119871169699128, −6.46890911572913046508142907506, −6.16614641491581033910791772439, −6.14733036677759785718725324436, −6.02759790741328477042840901861, −5.72312242033563260886396893140, −5.07546082215536363680178892796, −4.71668649373565350132669350449, −4.40614853134474044487183258207, −3.86890132869337120990603772110, −3.69765596098251923377977862788, −3.59585113682796791094767826017, −3.02171018969889954534999084204, −3.01883206035402986110805243031, −2.48722408296352792147509164224, −1.66612519306822245890503471034, −1.02962867128183891357801609789, −0.967239817515722642330527394836, −0.71211678185508057331020437424,
0.71211678185508057331020437424, 0.967239817515722642330527394836, 1.02962867128183891357801609789, 1.66612519306822245890503471034, 2.48722408296352792147509164224, 3.01883206035402986110805243031, 3.02171018969889954534999084204, 3.59585113682796791094767826017, 3.69765596098251923377977862788, 3.86890132869337120990603772110, 4.40614853134474044487183258207, 4.71668649373565350132669350449, 5.07546082215536363680178892796, 5.72312242033563260886396893140, 6.02759790741328477042840901861, 6.14733036677759785718725324436, 6.16614641491581033910791772439, 6.46890911572913046508142907506, 6.58995680535944119871169699128, 6.83050332634696054167295389856, 7.63988272991155122013862636542, 7.78457504866236935251638874729, 8.001510769350798589906393532469, 8.268849704843283699012750471437, 8.545552111989571715238918445272
