| L(s) = 1 | − 56·17-s + 4·25-s + 56·41-s − 92·49-s − 200·73-s − 248·89-s − 584·97-s − 520·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 292·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.29·17-s + 4/25·25-s + 1.36·41-s − 1.87·49-s − 2.73·73-s − 2.78·89-s − 6.02·97-s − 4.60·113-s − 3.20·121-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 + 1.72·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1402451229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1402451229\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1778 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1970 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3650 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 766 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1730 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4610 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 4370 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 866 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9506 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 12338 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13346 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 146 T + p^{2} T^{2} )^{4} \) |
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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
−6.81175359023397600793213001530, −6.79169715578140783084682174616, −6.38071637180955366143481168663, −6.25771656558400054110584313818, −5.76057407647167316015459949882, −5.74521201434080889466001090239, −5.60011908656237537009359215185, −5.15677882628717197904722718700, −4.95899227945489607510700670951, −4.77502362788234261874107951294, −4.38710679026083176891820221399, −4.12981683692882083309309991688, −4.09411953096704024660254226080, −4.07797998258258538752442754497, −3.60880956325498133923303033584, −2.89721639485615161859811581699, −2.87087492515470037426214589814, −2.73574769547300715286410537373, −2.62834116320571916426900803092, −1.96888665053320468111603082948, −1.63722128788802585918731814736, −1.63473733850677570280529901935, −1.15510574975976548763831012130, −0.48830889735201848498956170782, −0.07210398363851770413562992426,
0.07210398363851770413562992426, 0.48830889735201848498956170782, 1.15510574975976548763831012130, 1.63473733850677570280529901935, 1.63722128788802585918731814736, 1.96888665053320468111603082948, 2.62834116320571916426900803092, 2.73574769547300715286410537373, 2.87087492515470037426214589814, 2.89721639485615161859811581699, 3.60880956325498133923303033584, 4.07797998258258538752442754497, 4.09411953096704024660254226080, 4.12981683692882083309309991688, 4.38710679026083176891820221399, 4.77502362788234261874107951294, 4.95899227945489607510700670951, 5.15677882628717197904722718700, 5.60011908656237537009359215185, 5.74521201434080889466001090239, 5.76057407647167316015459949882, 6.25771656558400054110584313818, 6.38071637180955366143481168663, 6.79169715578140783084682174616, 6.81175359023397600793213001530
