| L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 3·9-s + 6·11-s + 2·13-s + 4·15-s + 2·19-s + 4·21-s − 4·23-s + 3·25-s + 4·27-s + 14·29-s + 4·31-s + 12·33-s + 4·35-s + 2·37-s + 4·39-s − 2·41-s − 6·43-s + 6·45-s − 12·47-s − 4·49-s + 8·53-s + 12·55-s + 4·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 1.80·11-s + 0.554·13-s + 1.03·15-s + 0.458·19-s + 0.872·21-s − 0.834·23-s + 3/5·25-s + 0.769·27-s + 2.59·29-s + 0.718·31-s + 2.08·33-s + 0.676·35-s + 0.328·37-s + 0.640·39-s − 0.312·41-s − 0.914·43-s + 0.894·45-s − 1.75·47-s − 4/7·49-s + 1.09·53-s + 1.61·55-s + 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.834309646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.834309646\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 14 T + 100 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 76 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 44 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.442396752394157986719624710511, −8.378309426815539897520853714466, −7.74741352423048493119381581866, −7.71332936677065818663580779064, −6.86459974175772536417860130436, −6.68380188688673030128796869597, −6.35845969825037222464822410375, −6.28668354697129778483231024563, −5.43414412069168753127154597723, −5.24226187610104879422397118322, −4.55045502353599921608921653401, −4.51807291121211765475816958586, −3.88111592719966943130413809287, −3.61543669127245862917558507276, −2.97344029339240830851385293651, −2.80265042955694524879887531705, −1.99921459068528355971127076153, −1.81118068223647028460996115851, −1.18900520423033290457766561742, −0.917318745951632453200851973070,
0.917318745951632453200851973070, 1.18900520423033290457766561742, 1.81118068223647028460996115851, 1.99921459068528355971127076153, 2.80265042955694524879887531705, 2.97344029339240830851385293651, 3.61543669127245862917558507276, 3.88111592719966943130413809287, 4.51807291121211765475816958586, 4.55045502353599921608921653401, 5.24226187610104879422397118322, 5.43414412069168753127154597723, 6.28668354697129778483231024563, 6.35845969825037222464822410375, 6.68380188688673030128796869597, 6.86459974175772536417860130436, 7.71332936677065818663580779064, 7.74741352423048493119381581866, 8.378309426815539897520853714466, 8.442396752394157986719624710511
