| L(s) = 1 | + 4·2-s + 6·4-s − 15·16-s + 4·23-s − 24·32-s + 16·46-s − 6·64-s + 4·79-s − 81-s + 24·92-s + 4·109-s + 4·113-s + 2·121-s + 127-s + 36·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·158-s − 4·162-s + 163-s + 167-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 4·2-s + 6·4-s − 15·16-s + 4·23-s − 24·32-s + 16·46-s − 6·64-s + 4·79-s − 81-s + 24·92-s + 4·109-s + 4·113-s + 2·121-s + 127-s + 36·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·158-s − 4·162-s + 163-s + 167-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(7.683224502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.683224502\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.05231875024656235219199305472, −5.82262556916373051868667095923, −5.70673311388995692299048142228, −5.43514013282420255398045784370, −5.41198716296725144750068194413, −5.05873013656178212325718798600, −4.84252052523841631714782420990, −4.76403826014019186605322703975, −4.73028675993986301333392718250, −4.56521418001281531156325148642, −4.36904262518443610069134769195, −4.04688947483348047150666125227, −3.83109395389377647197545016913, −3.54241573000367919136930449852, −3.40610627030842974818616456330, −3.19629348921094898149483370976, −3.13394024423283417938821040712, −3.12091039565285188317356420289, −2.67745929481078779548918971847, −2.28833442704790488035368851254, −2.18097009831759381127793093580, −2.03289039338714900240062601783, −1.20948653885162321301380129031, −0.74430272196132175298441590229, −0.73306197383431559847209943120,
0.73306197383431559847209943120, 0.74430272196132175298441590229, 1.20948653885162321301380129031, 2.03289039338714900240062601783, 2.18097009831759381127793093580, 2.28833442704790488035368851254, 2.67745929481078779548918971847, 3.12091039565285188317356420289, 3.13394024423283417938821040712, 3.19629348921094898149483370976, 3.40610627030842974818616456330, 3.54241573000367919136930449852, 3.83109395389377647197545016913, 4.04688947483348047150666125227, 4.36904262518443610069134769195, 4.56521418001281531156325148642, 4.73028675993986301333392718250, 4.76403826014019186605322703975, 4.84252052523841631714782420990, 5.05873013656178212325718798600, 5.41198716296725144750068194413, 5.43514013282420255398045784370, 5.70673311388995692299048142228, 5.82262556916373051868667095923, 6.05231875024656235219199305472
