| L(s) = 1 | − 176·25-s + 1.37e3·49-s + 2.36e3·73-s + 7.26e3·97-s + 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 1.40·25-s + 4·49-s + 3.79·73-s + 7.60·97-s + 4·121-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 + 3.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.693887746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.693887746\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2}( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 9776 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 36920 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )^{2}( 1 + 214 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 108560 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 296296 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 830 T + p^{3} T^{2} )^{2}( 1 + 830 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 592 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 293920 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1816 T + p^{3} T^{2} )^{4} \) |
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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
−6.74176298642129868513387874180, −6.25508070162468293524301652533, −6.10972274025756106884208621384, −5.99311497958081886867885834939, −5.90600983466363278497301777589, −5.26439620440131178641569128419, −5.24121891518899400473098676302, −5.21822942077086658545227330067, −4.92464325270578547466387448189, −4.43261357054831391304222712084, −4.13875098691493263615770527381, −4.09167988549421074766489940210, −4.02445352339883828502428147576, −3.49678427593935890537568747445, −3.26210936280549966826891035084, −3.10316283318569683037488345431, −2.87127513129457743335255197757, −2.18561971997690489414097174119, −2.11855601009482529338737617193, −1.97815831430881295318384661131, −1.88751110156751028046328377209, −1.05264432791191506259766879986, −0.72361445176203713162755789493, −0.66936967979856754033583082551, −0.42518359673694199852107123200,
0.42518359673694199852107123200, 0.66936967979856754033583082551, 0.72361445176203713162755789493, 1.05264432791191506259766879986, 1.88751110156751028046328377209, 1.97815831430881295318384661131, 2.11855601009482529338737617193, 2.18561971997690489414097174119, 2.87127513129457743335255197757, 3.10316283318569683037488345431, 3.26210936280549966826891035084, 3.49678427593935890537568747445, 4.02445352339883828502428147576, 4.09167988549421074766489940210, 4.13875098691493263615770527381, 4.43261357054831391304222712084, 4.92464325270578547466387448189, 5.21822942077086658545227330067, 5.24121891518899400473098676302, 5.26439620440131178641569128419, 5.90600983466363278497301777589, 5.99311497958081886867885834939, 6.10972274025756106884208621384, 6.25508070162468293524301652533, 6.74176298642129868513387874180
