| L(s) = 1 | + 2·5-s − 3·9-s − 4·11-s − 8·19-s + 5·25-s − 14·29-s + 12·31-s − 10·41-s − 6·45-s − 13·49-s − 8·55-s − 24·59-s + 14·61-s − 40·71-s + 8·79-s + 60·89-s − 16·95-s + 12·99-s − 36·101-s + 36·109-s + 26·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s − 28·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s − 1.20·11-s − 1.83·19-s + 25-s − 2.59·29-s + 2.15·31-s − 1.56·41-s − 0.894·45-s − 1.85·49-s − 1.07·55-s − 3.12·59-s + 1.79·61-s − 4.74·71-s + 0.900·79-s + 6.35·89-s − 1.64·95-s + 1.20·99-s − 3.58·101-s + 3.44·109-s + 2.36·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7835819283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7835819283\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 141 T^{2} + 12992 T^{4} + 141 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 62 T^{2} - 5565 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.446076383730760544529729604479, −8.060093109220541383188053156963, −7.71569621608747301890117549206, −7.65827593749543850263835344795, −7.47291762059091890183712370173, −6.93784266467325468788748321368, −6.79150902713689313699145238147, −6.29697638736949186263284442152, −6.23717542455689403995415937560, −6.06321869456946112507106623321, −5.89165328110235189086084430498, −5.40818015185060944672442689470, −5.12844368530982466486715151518, −4.87089503931260333837630675014, −4.80820835615152648658151524934, −4.27376311283525829727388139440, −4.11628447072577090996883996598, −3.42777209118915332564614815821, −3.25605554970502985338184883907, −2.98170143592387210645728165298, −2.62097249530466873784156637638, −2.05538764258521715343308577413, −1.97419577235365291095466726729, −1.46047502354722191220870568743, −0.33981968188771348506485971188,
0.33981968188771348506485971188, 1.46047502354722191220870568743, 1.97419577235365291095466726729, 2.05538764258521715343308577413, 2.62097249530466873784156637638, 2.98170143592387210645728165298, 3.25605554970502985338184883907, 3.42777209118915332564614815821, 4.11628447072577090996883996598, 4.27376311283525829727388139440, 4.80820835615152648658151524934, 4.87089503931260333837630675014, 5.12844368530982466486715151518, 5.40818015185060944672442689470, 5.89165328110235189086084430498, 6.06321869456946112507106623321, 6.23717542455689403995415937560, 6.29697638736949186263284442152, 6.79150902713689313699145238147, 6.93784266467325468788748321368, 7.47291762059091890183712370173, 7.65827593749543850263835344795, 7.71569621608747301890117549206, 8.060093109220541383188053156963, 8.446076383730760544529729604479
