| L(s) = 1 | − 2·5-s + 7-s + 3·11-s + 2·13-s − 3·17-s + 7·19-s − 3·23-s + 3·25-s + 3·29-s − 8·31-s − 2·35-s + 7·37-s − 9·41-s − 11·43-s + 7·49-s + 12·53-s − 6·55-s − 3·59-s − 11·61-s − 4·65-s + 7·67-s − 3·71-s + 4·73-s + 3·77-s + 16·79-s + 24·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s − 0.727·17-s + 1.60·19-s − 0.625·23-s + 3/5·25-s + 0.557·29-s − 1.43·31-s − 0.338·35-s + 1.15·37-s − 1.40·41-s − 1.67·43-s + 49-s + 1.64·53-s − 0.809·55-s − 0.390·59-s − 1.40·61-s − 0.496·65-s + 0.855·67-s − 0.356·71-s + 0.468·73-s + 0.341·77-s + 1.80·79-s + 2.63·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.279837739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.279837739\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.045335415708274092669804316075, −8.781573425291594366532997271134, −8.461652262778567891311077241473, −8.037229702763945028520136400748, −7.53682526483111914768791557216, −7.41367355618281310070987899494, −6.93305985241364853137976797836, −6.44494410167033123207761967230, −6.17360886011429375901506559812, −5.68779937876947023732891896898, −5.01748372213343689106150026376, −4.89388173357961108999736896143, −4.38677785551368905432978521657, −3.73908379344676860076380414262, −3.50178570023117928217447544442, −3.28549833200437993623794940665, −2.29278209313122599247025798979, −1.93127771079067251891264808421, −1.15359946678190835221702018690, −0.58893967528297735895479420427,
0.58893967528297735895479420427, 1.15359946678190835221702018690, 1.93127771079067251891264808421, 2.29278209313122599247025798979, 3.28549833200437993623794940665, 3.50178570023117928217447544442, 3.73908379344676860076380414262, 4.38677785551368905432978521657, 4.89388173357961108999736896143, 5.01748372213343689106150026376, 5.68779937876947023732891896898, 6.17360886011429375901506559812, 6.44494410167033123207761967230, 6.93305985241364853137976797836, 7.41367355618281310070987899494, 7.53682526483111914768791557216, 8.037229702763945028520136400748, 8.461652262778567891311077241473, 8.781573425291594366532997271134, 9.045335415708274092669804316075
