| L(s) = 1 | − 17·7-s − 89·13-s + 214·19-s + 125·25-s − 308·31-s − 866·37-s + 520·43-s + 343·49-s + 901·61-s − 1.00e3·67-s − 542·73-s − 503·79-s + 1.51e3·91-s − 1.85e3·97-s + 19·103-s − 1.29e3·109-s + 1.33e3·121-s + 127-s + 131-s − 3.63e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.917·7-s − 1.89·13-s + 2.58·19-s + 25-s − 1.78·31-s − 3.84·37-s + 1.84·43-s + 49-s + 1.89·61-s − 1.83·67-s − 0.868·73-s − 0.716·79-s + 1.74·91-s − 1.93·97-s + 0.0181·103-s − 1.13·109-s + 121-s + 0.000698·127-s + 0.000666·131-s − 2.37·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.091497212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.091497212\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )( 1 + 37 T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )( 1 + 70 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 107 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )( 1 + 289 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 433 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 449 T + p^{3} T^{2} )( 1 - 71 T + p^{3} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 719 T + p^{3} T^{2} )( 1 - 182 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 127 T + p^{3} T^{2} )( 1 + 880 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 271 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )( 1 + 1387 T + p^{3} T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 523 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \) |
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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
−11.49380393827533142355984799335, −10.83481079229787325935131551386, −10.44787895345491171515221873567, −9.955576646757504687975032636207, −9.576388470309955485764095959695, −9.144392188729630185390396760906, −8.812143734218128454249921915932, −8.036583517604086068262059213962, −7.18609054441951655923459920399, −7.15276909921579537655604378267, −7.01958430435382091365329986772, −5.80002858783243465787826093162, −5.42190152205182772722049638806, −5.12903352169656611318515130536, −4.31691669560371039256019514301, −3.44595979829924927111067781997, −3.12972756338506553802889723536, −2.39704149985424468385239901056, −1.45428318178761206728967840321, −0.38151929726089190924900171525,
0.38151929726089190924900171525, 1.45428318178761206728967840321, 2.39704149985424468385239901056, 3.12972756338506553802889723536, 3.44595979829924927111067781997, 4.31691669560371039256019514301, 5.12903352169656611318515130536, 5.42190152205182772722049638806, 5.80002858783243465787826093162, 7.01958430435382091365329986772, 7.15276909921579537655604378267, 7.18609054441951655923459920399, 8.036583517604086068262059213962, 8.812143734218128454249921915932, 9.144392188729630185390396760906, 9.576388470309955485764095959695, 9.955576646757504687975032636207, 10.44787895345491171515221873567, 10.83481079229787325935131551386, 11.49380393827533142355984799335
