| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s + 2·11-s + 5·16-s + 6·17-s − 2·19-s + 6·20-s − 4·22-s − 4·23-s − 2·25-s − 10·31-s − 6·32-s − 12·34-s − 4·37-s + 4·38-s − 8·40-s + 6·41-s + 4·43-s + 6·44-s + 8·46-s + 6·47-s + 4·50-s + 12·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 1.26·10-s + 0.603·11-s + 5/4·16-s + 1.45·17-s − 0.458·19-s + 1.34·20-s − 0.852·22-s − 0.834·23-s − 2/5·25-s − 1.79·31-s − 1.06·32-s − 2.05·34-s − 0.657·37-s + 0.648·38-s − 1.26·40-s + 0.937·41-s + 0.609·43-s + 0.904·44-s + 1.17·46-s + 0.875·47-s + 0.565·50-s + 1.64·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131124006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131124006\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 162 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−7.64122486045090032868752395331, −7.51094990459684057901788150817, −7.23467212132357021349930817359, −7.21406646171574540605324781367, −6.42112028250965729792088149889, −6.07815469823933489417662159348, −5.94958910523081207352979633510, −5.76848086573341408127054009480, −5.17214062048125914071591313762, −4.98932281762874369630064853677, −4.16682813317038650899691498016, −3.93731539192904950324502765614, −3.54576248318191699989054317023, −3.18832118192243403912499352078, −2.42480316453306053751545513846, −2.36041988340953305810774911917, −1.68590016565790179826447577587, −1.64100581346273029575897509281, −0.816454758009094986539579143561, −0.52139969252379929261967267725,
0.52139969252379929261967267725, 0.816454758009094986539579143561, 1.64100581346273029575897509281, 1.68590016565790179826447577587, 2.36041988340953305810774911917, 2.42480316453306053751545513846, 3.18832118192243403912499352078, 3.54576248318191699989054317023, 3.93731539192904950324502765614, 4.16682813317038650899691498016, 4.98932281762874369630064853677, 5.17214062048125914071591313762, 5.76848086573341408127054009480, 5.94958910523081207352979633510, 6.07815469823933489417662159348, 6.42112028250965729792088149889, 7.21406646171574540605324781367, 7.23467212132357021349930817359, 7.51094990459684057901788150817, 7.64122486045090032868752395331
