| L(s) = 1 | + 2·9-s + 2·11-s − 14·19-s − 6·29-s − 10·31-s + 24·41-s − 2·49-s + 24·59-s − 20·61-s − 6·71-s − 8·79-s − 5·81-s − 6·89-s + 4·99-s − 18·101-s − 10·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 0.603·11-s − 3.21·19-s − 1.11·29-s − 1.79·31-s + 3.74·41-s − 2/7·49-s + 3.12·59-s − 2.56·61-s − 0.712·71-s − 0.900·79-s − 5/9·81-s − 0.635·89-s + 0.402·99-s − 1.79·101-s − 0.957·109-s + 3/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.046466160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046466160\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−8.926217717685281838892111830614, −7.981394228156935909087808489336, −7.914734302062534816829418590942, −7.48871966016071876295009338258, −7.02317225800369738461371749946, −6.70402907059729883242838634684, −6.47708827817273426235087376681, −5.93143369643848377973489483197, −5.63462800973695611553518585866, −5.41543068062785251297950794509, −4.55851321986558276949923268078, −4.29129494815782552133017170789, −4.08249705499644568969030083261, −3.85852787700868247762077957963, −3.15812745011950055160925598309, −2.42777646427204561776212830375, −2.31292953160749117683374370480, −1.67115735044184490312501231863, −1.25017265264087383833378319018, −0.28494925438903173039238811988,
0.28494925438903173039238811988, 1.25017265264087383833378319018, 1.67115735044184490312501231863, 2.31292953160749117683374370480, 2.42777646427204561776212830375, 3.15812745011950055160925598309, 3.85852787700868247762077957963, 4.08249705499644568969030083261, 4.29129494815782552133017170789, 4.55851321986558276949923268078, 5.41543068062785251297950794509, 5.63462800973695611553518585866, 5.93143369643848377973489483197, 6.47708827817273426235087376681, 6.70402907059729883242838634684, 7.02317225800369738461371749946, 7.48871966016071876295009338258, 7.914734302062534816829418590942, 7.981394228156935909087808489336, 8.926217717685281838892111830614
