| L(s) = 1 | + 2·4-s + 32·13-s + 32·19-s − 32·25-s − 88·31-s + 68·37-s − 160·43-s + 64·52-s − 100·61-s − 8·64-s − 16·67-s + 32·73-s + 64·76-s + 152·79-s + 704·97-s − 64·100-s + 56·103-s − 112·109-s + 46·121-s − 176·124-s + 127-s + 131-s + 137-s + 139-s + 136·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2.46·13-s + 1.68·19-s − 1.27·25-s − 2.83·31-s + 1.83·37-s − 3.72·43-s + 1.23·52-s − 1.63·61-s − 1/8·64-s − 0.238·67-s + 0.438·73-s + 0.842·76-s + 1.92·79-s + 7.25·97-s − 0.639·100-s + 0.543·103-s − 1.02·109-s + 0.380·121-s − 1.41·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.918·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.001212948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.001212948\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 416 T^{2} + 89535 T^{4} + 416 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 16 T - 105 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1664 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 44 T + 975 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T - 213 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 2782 T^{2} + 2859843 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 4160 T^{2} + 9415119 T^{4} + 4160 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 5810 T^{2} + 21638739 T^{4} + 5810 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 50 T - 1221 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T - 4425 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 16 T - 5073 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 76 T - 465 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 15680 T^{2} + 183120159 T^{4} + 15680 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 176 T + p^{2} T^{2} )^{4} \) |
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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
−7.29510597043918513824478881988, −6.68960507114634937417583612715, −6.58454296442981762452584461267, −6.42655846622703584232587513997, −6.08827357587359628402551617166, −5.79926872273733297714545870604, −5.79526518098685410539451085865, −5.70977104975185188391608674850, −5.17620082455211142542757654344, −4.86489606911471324929308752611, −4.86261662107052543545178495424, −4.42394723127628673032919476038, −4.24448905908103766497422397014, −3.67209649549636107384458131348, −3.49185881912232052415658736361, −3.41923875675352544471554605053, −3.25326041481633603663054732158, −3.16231261535100966756269436107, −2.20146259006655249048725598261, −2.08691683668623345723881941881, −2.02502157391267709299706532480, −1.50739936844273005844238514381, −1.11392023832488626151105945597, −0.895555127605426677625014775021, −0.25185809599347690812056891714,
0.25185809599347690812056891714, 0.895555127605426677625014775021, 1.11392023832488626151105945597, 1.50739936844273005844238514381, 2.02502157391267709299706532480, 2.08691683668623345723881941881, 2.20146259006655249048725598261, 3.16231261535100966756269436107, 3.25326041481633603663054732158, 3.41923875675352544471554605053, 3.49185881912232052415658736361, 3.67209649549636107384458131348, 4.24448905908103766497422397014, 4.42394723127628673032919476038, 4.86261662107052543545178495424, 4.86489606911471324929308752611, 5.17620082455211142542757654344, 5.70977104975185188391608674850, 5.79526518098685410539451085865, 5.79926872273733297714545870604, 6.08827357587359628402551617166, 6.42655846622703584232587513997, 6.58454296442981762452584461267, 6.68960507114634937417583612715, 7.29510597043918513824478881988
