| L(s) = 1 | − 14·5-s − 22·9-s + 40·11-s + 168·19-s + 71·25-s + 12·29-s + 448·31-s − 532·41-s + 308·45-s − 560·55-s + 56·59-s − 364·61-s + 816·71-s + 96·79-s − 245·81-s + 3.05e3·89-s − 2.35e3·95-s − 880·99-s − 2.49e3·101-s + 1.80e3·109-s − 1.46e3·121-s + 756·125-s + 127-s + 131-s + 137-s + 139-s − 168·145-s + ⋯ |
| L(s) = 1 | − 1.25·5-s − 0.814·9-s + 1.09·11-s + 2.02·19-s + 0.567·25-s + 0.0768·29-s + 2.59·31-s − 2.02·41-s + 1.02·45-s − 1.37·55-s + 0.123·59-s − 0.764·61-s + 1.36·71-s + 0.136·79-s − 0.336·81-s + 3.63·89-s − 2.54·95-s − 0.893·99-s − 2.45·101-s + 1.58·109-s − 1.09·121-s + 0.540·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.0962·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.084762517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084762517\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + 14 T + p^{3} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 22 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1658 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4962 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 84 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )( 1 + 212 T + p^{3} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 86410 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 65914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 67122 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 163690 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 419050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 408 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 390542 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1103370 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1526 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1514050 T^{2} + p^{6} T^{4} \) |
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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
−9.894419570481296276460799943866, −9.182615061368038213817505568919, −9.142725488969848189902819096467, −8.416752559664561233223824510673, −8.220491696296920183398601980657, −7.75866037548055847635655362391, −7.42934797799590261162254203845, −6.85727493164325018002253492761, −6.36573273189572536023011542747, −6.22064243607079305175184975081, −5.29194501391218852563580586989, −5.07212168482786134513381667924, −4.58187374163327440971283046010, −3.80932730252382022472053994343, −3.67101769386755816820557424106, −3.00282343233234447645397530519, −2.66762321550866728266639239838, −1.61757381648431326955826081876, −0.992594452093018457850202554295, −0.44879996892751431041188353600,
0.44879996892751431041188353600, 0.992594452093018457850202554295, 1.61757381648431326955826081876, 2.66762321550866728266639239838, 3.00282343233234447645397530519, 3.67101769386755816820557424106, 3.80932730252382022472053994343, 4.58187374163327440971283046010, 5.07212168482786134513381667924, 5.29194501391218852563580586989, 6.22064243607079305175184975081, 6.36573273189572536023011542747, 6.85727493164325018002253492761, 7.42934797799590261162254203845, 7.75866037548055847635655362391, 8.220491696296920183398601980657, 8.416752559664561233223824510673, 9.142725488969848189902819096467, 9.182615061368038213817505568919, 9.894419570481296276460799943866
