| L(s) = 1 | − 2·5-s + 2·7-s + 2·11-s − 10·13-s + 12·17-s − 2·19-s + 2·23-s + 3·25-s − 4·29-s − 2·31-s − 4·35-s − 2·37-s + 12·41-s + 2·47-s + 49-s + 4·53-s − 4·55-s + 4·59-s − 18·61-s + 20·65-s + 10·67-s + 4·71-s − 26·73-s + 4·77-s + 8·79-s − 4·83-s − 24·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 0.603·11-s − 2.77·13-s + 2.91·17-s − 0.458·19-s + 0.417·23-s + 3/5·25-s − 0.742·29-s − 0.359·31-s − 0.676·35-s − 0.328·37-s + 1.87·41-s + 0.291·47-s + 1/7·49-s + 0.549·53-s − 0.539·55-s + 0.520·59-s − 2.30·61-s + 2.48·65-s + 1.22·67-s + 0.474·71-s − 3.04·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s − 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.339492237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.339492237\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 115 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 95 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 200 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 26 T + 312 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 95 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−8.398394972117492602225408327999, −8.271666437361429482486957972676, −7.55491602449439102140022244576, −7.48471910346565190155131549742, −7.31013820306469509939203608450, −7.25851005391246496524177318065, −6.22161442695359963300102650094, −6.11824043870575387337461853206, −5.42390838381168569674357040944, −5.31749171100122395914654245978, −4.80811105984273144911628066603, −4.47178189430518470967501179811, −4.17806601028777823618010976529, −3.53442423262008260350546858260, −3.15549718158090759245168180559, −2.87801811757684074313834299640, −2.07921534633835402326857337660, −1.84670153597132589616586768874, −0.944740134060137521817513579406, −0.53604838269124823699051315911,
0.53604838269124823699051315911, 0.944740134060137521817513579406, 1.84670153597132589616586768874, 2.07921534633835402326857337660, 2.87801811757684074313834299640, 3.15549718158090759245168180559, 3.53442423262008260350546858260, 4.17806601028777823618010976529, 4.47178189430518470967501179811, 4.80811105984273144911628066603, 5.31749171100122395914654245978, 5.42390838381168569674357040944, 6.11824043870575387337461853206, 6.22161442695359963300102650094, 7.25851005391246496524177318065, 7.31013820306469509939203608450, 7.48471910346565190155131549742, 7.55491602449439102140022244576, 8.271666437361429482486957972676, 8.398394972117492602225408327999
