| L(s) = 1 | + 3-s + 3·5-s + 3·9-s + 8·11-s − 5·13-s + 3·15-s + 5·17-s − 8·19-s + 23-s + 5·25-s + 8·27-s + 3·29-s + 8·31-s + 8·33-s − 4·37-s − 5·39-s + 5·41-s − 11·43-s + 9·45-s + 5·47-s − 14·49-s + 5·51-s − 9·53-s + 24·55-s − 8·57-s + 13·59-s − 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 9-s + 2.41·11-s − 1.38·13-s + 0.774·15-s + 1.21·17-s − 1.83·19-s + 0.208·23-s + 25-s + 1.53·27-s + 0.557·29-s + 1.43·31-s + 1.39·33-s − 0.657·37-s − 0.800·39-s + 0.780·41-s − 1.67·43-s + 1.34·45-s + 0.729·47-s − 2·49-s + 0.700·51-s − 1.23·53-s + 3.23·55-s − 1.05·57-s + 1.69·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.680950561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.680950561\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.09218622119851267971677131830, −9.623336740363681090136830860476, −9.185990198395943750064219674179, −8.618370361336913708281870799250, −8.577284145241727260502725660451, −8.047099642116034488546395432749, −7.16037949655193162309034997164, −7.13509691775498423330180611495, −6.59867913544136811142849482496, −6.19763187865265040010956589756, −6.03593189324504591406776830332, −5.16188938104293955838932428878, −4.62288120007408061392660709754, −4.49860334420986502597813157110, −3.85058148625644840414696704793, −3.16848134761799190016259149630, −2.76958147167755201392029308328, −1.92866026723187867845631256551, −1.65343395986600288155362959938, −0.973127705026115125339362395937,
0.973127705026115125339362395937, 1.65343395986600288155362959938, 1.92866026723187867845631256551, 2.76958147167755201392029308328, 3.16848134761799190016259149630, 3.85058148625644840414696704793, 4.49860334420986502597813157110, 4.62288120007408061392660709754, 5.16188938104293955838932428878, 6.03593189324504591406776830332, 6.19763187865265040010956589756, 6.59867913544136811142849482496, 7.13509691775498423330180611495, 7.16037949655193162309034997164, 8.047099642116034488546395432749, 8.577284145241727260502725660451, 8.618370361336913708281870799250, 9.185990198395943750064219674179, 9.623336740363681090136830860476, 10.09218622119851267971677131830
