| L(s) = 1 | + 3·3-s + 3·7-s + 4·9-s + 7·11-s − 3·13-s − 3·17-s − 19-s + 9·21-s − 2·23-s + 6·27-s + 2·29-s + 5·31-s + 21·33-s − 16·37-s − 9·39-s − 9·41-s − 4·43-s + 2·47-s − 4·49-s − 9·51-s + 8·53-s − 3·57-s + 14·59-s + 5·61-s + 12·63-s − 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.13·7-s + 4/3·9-s + 2.11·11-s − 0.832·13-s − 0.727·17-s − 0.229·19-s + 1.96·21-s − 0.417·23-s + 1.15·27-s + 0.371·29-s + 0.898·31-s + 3.65·33-s − 2.63·37-s − 1.44·39-s − 1.40·41-s − 0.609·43-s + 0.291·47-s − 4/7·49-s − 1.26·51-s + 1.09·53-s − 0.397·57-s + 1.82·59-s + 0.640·61-s + 1.51·63-s − 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.075416071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.075416071\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 29 T + 349 T^{2} - 29 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 211 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.897761625377557783958830302705, −7.80577908841830851465770637617, −7.12372853348277097486384293938, −6.74951995587140740420234874841, −6.72305917635258793596613043464, −6.61669080313727918408999812333, −5.67290259234588274278594436600, −5.53922183518183520686794501609, −4.89786819222153260394355570995, −4.68863635319993360633080876752, −4.41764507930794717910725229365, −3.88170565111626186677213893975, −3.52843420231793789519017948575, −3.41939418655093580837866840198, −2.82667419385472533875207676674, −2.24049080207879389682714202645, −2.06179785899370533430449768371, −1.70514188051793267288784847111, −1.21038784075603931654064442333, −0.55261817108703042646274287916,
0.55261817108703042646274287916, 1.21038784075603931654064442333, 1.70514188051793267288784847111, 2.06179785899370533430449768371, 2.24049080207879389682714202645, 2.82667419385472533875207676674, 3.41939418655093580837866840198, 3.52843420231793789519017948575, 3.88170565111626186677213893975, 4.41764507930794717910725229365, 4.68863635319993360633080876752, 4.89786819222153260394355570995, 5.53922183518183520686794501609, 5.67290259234588274278594436600, 6.61669080313727918408999812333, 6.72305917635258793596613043464, 6.74951995587140740420234874841, 7.12372853348277097486384293938, 7.80577908841830851465770637617, 7.897761625377557783958830302705
