| L(s) = 1 | + 2·5-s + 9-s + 20·11-s − 14·19-s + 5·25-s + 2·29-s + 12·31-s + 12·41-s + 2·45-s + 28·49-s + 40·55-s + 22·59-s + 6·61-s + 18·71-s − 10·79-s + 2·89-s − 28·95-s + 20·99-s + 18·101-s − 10·109-s + 206·121-s + 22·125-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s + 6.03·11-s − 3.21·19-s + 25-s + 0.371·29-s + 2.15·31-s + 1.87·41-s + 0.298·45-s + 4·49-s + 5.39·55-s + 2.86·59-s + 0.768·61-s + 2.13·71-s − 1.12·79-s + 0.211·89-s − 2.87·95-s + 2.01·99-s + 1.79·101-s − 0.957·109-s + 18.7·121-s + 1.96·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.30878035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.30878035\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 50 T^{2} + 651 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.12499597536177351128413821857, −6.63167526293814998909447949033, −6.53200965453827543655334881932, −6.32058511204070234381946496722, −6.25082629320734685420503656861, −6.10063700517006539258325503425, −5.82695454279804881233042705615, −5.71604382527117879744619288853, −5.17497290731132937610946724215, −4.72584534851249827071511508897, −4.67692873369569731273650588750, −4.34160093556515063166930123437, −4.31576784133966446484652567151, −3.82475595931338171569167716182, −3.72996926623664892572391886976, −3.69226678641939762942076269814, −3.67420783998472470280164858567, −2.68569501076361969654380950068, −2.35826849230402189882295165130, −2.31672108584969023587101651781, −2.20386782995372690811850964012, −1.43478807967066484004565850249, −1.21732075667350323593644017219, −1.05377615483316965190485585354, −0.821480638057908420670124312216,
0.821480638057908420670124312216, 1.05377615483316965190485585354, 1.21732075667350323593644017219, 1.43478807967066484004565850249, 2.20386782995372690811850964012, 2.31672108584969023587101651781, 2.35826849230402189882295165130, 2.68569501076361969654380950068, 3.67420783998472470280164858567, 3.69226678641939762942076269814, 3.72996926623664892572391886976, 3.82475595931338171569167716182, 4.31576784133966446484652567151, 4.34160093556515063166930123437, 4.67692873369569731273650588750, 4.72584534851249827071511508897, 5.17497290731132937610946724215, 5.71604382527117879744619288853, 5.82695454279804881233042705615, 6.10063700517006539258325503425, 6.25082629320734685420503656861, 6.32058511204070234381946496722, 6.53200965453827543655334881932, 6.63167526293814998909447949033, 7.12499597536177351128413821857
