| L(s) = 1 | − 3-s − 2·5-s + 9-s + 8·11-s + 2·15-s + 4·17-s − 25-s − 27-s − 8·33-s + 8·43-s − 2·45-s − 14·49-s − 4·51-s − 4·53-s − 16·55-s + 8·59-s − 4·61-s − 8·67-s + 16·71-s + 75-s + 81-s − 8·85-s + 8·99-s + 32·103-s − 4·109-s + 36·113-s + 26·121-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 2.41·11-s + 0.516·15-s + 0.970·17-s − 1/5·25-s − 0.192·27-s − 1.39·33-s + 1.21·43-s − 0.298·45-s − 2·49-s − 0.560·51-s − 0.549·53-s − 2.15·55-s + 1.04·59-s − 0.512·61-s − 0.977·67-s + 1.89·71-s + 0.115·75-s + 1/9·81-s − 0.867·85-s + 0.804·99-s + 3.15·103-s − 0.383·109-s + 3.38·113-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.200769361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.200769361\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.811440441480316702575338596556, −9.117637452350560865270424633223, −8.825846092924041988502122678303, −8.098990694093691505068092710868, −7.62196275802298861591436072358, −7.18888718297647721192763594914, −6.42897107072744719896577758454, −6.30978253779542156728101297956, −5.57054830111212428433722417563, −4.79739013158573068367880494367, −4.25303028692796488061253338187, −3.70365087696193807260970500510, −3.25678719031186811580504077978, −1.85007062551041405309421470085, −0.935452235825979143826970020046,
0.935452235825979143826970020046, 1.85007062551041405309421470085, 3.25678719031186811580504077978, 3.70365087696193807260970500510, 4.25303028692796488061253338187, 4.79739013158573068367880494367, 5.57054830111212428433722417563, 6.30978253779542156728101297956, 6.42897107072744719896577758454, 7.18888718297647721192763594914, 7.62196275802298861591436072358, 8.098990694093691505068092710868, 8.825846092924041988502122678303, 9.117637452350560865270424633223, 9.811440441480316702575338596556
