| L(s) = 1 | − 4·3-s + 20·5-s − 36·7-s + 8·9-s − 66·13-s − 80·15-s + 134·17-s − 232·19-s + 144·21-s + 220·23-s + 275·25-s − 108·27-s − 720·35-s + 130·37-s + 264·39-s − 608·41-s − 308·43-s + 160·45-s + 612·47-s + 648·49-s − 536·51-s + 434·53-s + 928·57-s + 408·59-s − 1.49e3·61-s − 288·63-s − 1.32e3·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 1.78·5-s − 1.94·7-s + 8/27·9-s − 1.40·13-s − 1.37·15-s + 1.91·17-s − 2.80·19-s + 1.49·21-s + 1.99·23-s + 11/5·25-s − 0.769·27-s − 3.47·35-s + 0.577·37-s + 1.08·39-s − 2.31·41-s − 1.09·43-s + 0.530·45-s + 1.89·47-s + 1.88·49-s − 1.47·51-s + 1.12·53-s + 2.15·57-s + 0.900·59-s − 3.14·61-s − 0.575·63-s − 2.51·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.231055972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.231055972\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2406 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 66 T + 2178 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 134 T + 8978 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 116 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 220 T + 24200 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 31354 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 54958 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 130 T + 8450 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 304 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 308 T + 47432 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 612 T + 187272 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 434 T + 94178 T^{2} - 434 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 204 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 748 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 332 T + 55112 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 441246 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 554 T + 153458 T^{2} - 554 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1232 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 580 T + 168200 T^{2} - 580 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 769938 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1302 T + 847602 T^{2} + 1302 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.54777526963283778414428458798, −12.34749596978155582594518110271, −12.01591923968276918935703455184, −10.79311913473788208104889868637, −10.56715084971419054908330000057, −10.21900252284290068286482143291, −9.612659703720616504281396489938, −9.388859466764214032809170128021, −8.866086659942218798357903936377, −7.975786781302583943782174290379, −7.00433597116835647010811293373, −6.67276795358821284924983711230, −6.34643420035326046729514488097, −5.53954638557897931216455851357, −5.36582194795881640575977434363, −4.46292801807771220142490614186, −3.33416537283852494718440498644, −2.71645209377501808251041458927, −1.86251808771981867921032327061, −0.53163146395885567769562470827,
0.53163146395885567769562470827, 1.86251808771981867921032327061, 2.71645209377501808251041458927, 3.33416537283852494718440498644, 4.46292801807771220142490614186, 5.36582194795881640575977434363, 5.53954638557897931216455851357, 6.34643420035326046729514488097, 6.67276795358821284924983711230, 7.00433597116835647010811293373, 7.975786781302583943782174290379, 8.866086659942218798357903936377, 9.388859466764214032809170128021, 9.612659703720616504281396489938, 10.21900252284290068286482143291, 10.56715084971419054908330000057, 10.79311913473788208104889868637, 12.01591923968276918935703455184, 12.34749596978155582594518110271, 12.54777526963283778414428458798
