L(s) = 1 | + 2·2-s + 3·3-s + 3·4-s + 5-s + 6·6-s + 4·8-s + 6·9-s + 2·10-s + 2·11-s + 9·12-s + 2·13-s + 3·15-s + 5·16-s + 12·18-s − 7·19-s + 3·20-s + 4·22-s − 3·23-s + 12·24-s + 5·25-s + 4·26-s + 9·27-s + 8·29-s + 6·30-s − 8·31-s + 6·32-s + 6·33-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.447·5-s + 2.44·6-s + 1.41·8-s + 2·9-s + 0.632·10-s + 0.603·11-s + 2.59·12-s + 0.554·13-s + 0.774·15-s + 5/4·16-s + 2.82·18-s − 1.60·19-s + 0.670·20-s + 0.852·22-s − 0.625·23-s + 2.44·24-s + 25-s + 0.784·26-s + 1.73·27-s + 1.48·29-s + 1.09·30-s − 1.43·31-s + 1.06·32-s + 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.50040835\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.50040835\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.24928735491817010990957004318, −10.20146749885659175990741130290, −9.198450010536496057141673274143, −9.080708600348825569685490039945, −8.763741754387745562606186147565, −8.278154589518946319447741292519, −7.57213595954818906173831958355, −7.53520605461868283020667092820, −6.81944051310264722369827652064, −6.41835757670828001597970539845, −6.02099304768654761605733423058, −5.63479974483130136865129532210, −4.60337356946091157326254310396, −4.52277536523174472076842759039, −4.03548871328453566246913551667, −3.51325462622673589719819883675, −2.80455990106201792401222549283, −2.71671479382720217630510662210, −1.77528803446710747546397883838, −1.47567785996464988209168649710,
1.47567785996464988209168649710, 1.77528803446710747546397883838, 2.71671479382720217630510662210, 2.80455990106201792401222549283, 3.51325462622673589719819883675, 4.03548871328453566246913551667, 4.52277536523174472076842759039, 4.60337356946091157326254310396, 5.63479974483130136865129532210, 6.02099304768654761605733423058, 6.41835757670828001597970539845, 6.81944051310264722369827652064, 7.53520605461868283020667092820, 7.57213595954818906173831958355, 8.278154589518946319447741292519, 8.763741754387745562606186147565, 9.080708600348825569685490039945, 9.198450010536496057141673274143, 10.20146749885659175990741130290, 10.24928735491817010990957004318