L(s) = 1 | − 8·5-s − 4·16-s − 4·23-s + 38·25-s − 10·31-s + 6·37-s − 4·47-s − 13·49-s − 12·53-s + 20·59-s − 2·67-s + 32·80-s − 24·89-s + 10·97-s − 14·103-s − 12·113-s + 32·115-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 80·155-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 3.57·5-s − 16-s − 0.834·23-s + 38/5·25-s − 1.79·31-s + 0.986·37-s − 0.583·47-s − 1.85·49-s − 1.64·53-s + 2.60·59-s − 0.244·67-s + 3.57·80-s − 2.54·89-s + 1.01·97-s − 1.37·103-s − 1.12·113-s + 2.98·115-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 6.42·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−7.58382685421445972557056040132, −7.28899889706555778737663428863, −6.92262314684549227272101394731, −6.56051958738602663681229923169, −5.84618451955771691498419677236, −5.05774622025620960162902603755, −4.75210585764513502454372949525, −4.22709146321447807396748070156, −3.93444342612617848308073041065, −3.52742564992718846698837310315, −3.09166168739687900450908810991, −2.34720842629440559581056516038, −1.23603098910894570836676745841, 0, 0,
1.23603098910894570836676745841, 2.34720842629440559581056516038, 3.09166168739687900450908810991, 3.52742564992718846698837310315, 3.93444342612617848308073041065, 4.22709146321447807396748070156, 4.75210585764513502454372949525, 5.05774622025620960162902603755, 5.84618451955771691498419677236, 6.56051958738602663681229923169, 6.92262314684549227272101394731, 7.28899889706555778737663428863, 7.58382685421445972557056040132