| L(s) = 1 | + 3·3-s − 5·7-s + 6·9-s − 15·19-s − 15·21-s + 5·25-s + 9·27-s − 3·31-s + 11·37-s + 26·43-s + 18·49-s − 45·57-s − 12·61-s − 30·63-s − 5·67-s − 3·73-s + 15·75-s − 17·79-s + 9·81-s − 9·93-s + 27·103-s − 19·109-s + 33·111-s − 11·121-s + 127-s + 78·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.88·7-s + 2·9-s − 3.44·19-s − 3.27·21-s + 25-s + 1.73·27-s − 0.538·31-s + 1.80·37-s + 3.96·43-s + 18/7·49-s − 5.96·57-s − 1.53·61-s − 3.77·63-s − 0.610·67-s − 0.351·73-s + 1.73·75-s − 1.91·79-s + 81-s − 0.933·93-s + 2.66·103-s − 1.81·109-s + 3.13·111-s − 121-s + 0.0887·127-s + 6.86·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.271615146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271615146\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−14.45671112541079552165042467063, −14.19403850596027687517940994606, −13.22231233716616750026509569343, −13.10525951925068965063688358972, −12.57254959801065830060291352320, −12.43937190505582300927148332302, −11.00061518185197420179823481078, −10.61244153110935554435975147557, −10.05107202426334850739651401692, −9.371346798397679173883712076492, −8.863905064716110456257889432496, −8.733724148116538139149901003415, −7.70081156292783400695111500847, −7.23827633619799554746052242284, −6.30791160156048248823933360980, −6.10574114292596037424511130778, −4.25090263617993966958778313709, −4.08732909027933863056444995936, −2.87999174247487663390578797746, −2.41433171051343936092513682492,
2.41433171051343936092513682492, 2.87999174247487663390578797746, 4.08732909027933863056444995936, 4.25090263617993966958778313709, 6.10574114292596037424511130778, 6.30791160156048248823933360980, 7.23827633619799554746052242284, 7.70081156292783400695111500847, 8.733724148116538139149901003415, 8.863905064716110456257889432496, 9.371346798397679173883712076492, 10.05107202426334850739651401692, 10.61244153110935554435975147557, 11.00061518185197420179823481078, 12.43937190505582300927148332302, 12.57254959801065830060291352320, 13.10525951925068965063688358972, 13.22231233716616750026509569343, 14.19403850596027687517940994606, 14.45671112541079552165042467063
