| L(s) = 1 | − 2·9-s − 25-s − 4·29-s + 3·81-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + 227-s + ⋯ |
| L(s) = 1 | − 2·9-s − 25-s − 4·29-s + 3·81-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5680167293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5680167293\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.839457742917624730236885057138, −8.368707523433423459779826700280, −8.288877930214538820440036348875, −7.53641197172638003296870852757, −7.47101949045727111959641763989, −7.26430085416523957860440244634, −6.46017603534259207123998851816, −6.08527879990415213721264593576, −5.87953827448520292808163206630, −5.60599793423335018263898031548, −5.13493417296452068192629447385, −4.87260657342359490919098493428, −4.13251387569202791887804554781, −3.72727829830711425869347741413, −3.44228616225946584095229912673, −3.06315498666256162455777069828, −2.24684707486381082344021250918, −2.19577350563495235354461058585, −1.52752880183226603916270052959, −0.40325904393571257818145691232,
0.40325904393571257818145691232, 1.52752880183226603916270052959, 2.19577350563495235354461058585, 2.24684707486381082344021250918, 3.06315498666256162455777069828, 3.44228616225946584095229912673, 3.72727829830711425869347741413, 4.13251387569202791887804554781, 4.87260657342359490919098493428, 5.13493417296452068192629447385, 5.60599793423335018263898031548, 5.87953827448520292808163206630, 6.08527879990415213721264593576, 6.46017603534259207123998851816, 7.26430085416523957860440244634, 7.47101949045727111959641763989, 7.53641197172638003296870852757, 8.288877930214538820440036348875, 8.368707523433423459779826700280, 8.839457742917624730236885057138
