| L(s) = 1 | − 27·9-s + 140·13-s + 250·25-s + 220·37-s − 286·49-s + 364·61-s − 2.38e3·73-s + 729·81-s + 2.66e3·97-s + 1.29e3·109-s − 3.78e3·117-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s + 2.98·13-s + 2·25-s + 0.977·37-s − 0.833·49-s + 0.764·61-s − 3.81·73-s + 81-s + 2.78·97-s + 1.13·109-s − 2.98·117-s − 2·121-s + 0.000698·127-s + 0.000666·131-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 + 4.69·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.688077762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688077762\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )( 1 + 20 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )( 1 + 56 T + p^{3} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 110 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )( 1 + 520 T + p^{3} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )( 1 + 880 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1190 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )( 1 + 884 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )^{2} \) |
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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
−15.57247524311043940173259685686, −14.64698277918031551293222793826, −14.48397440827174773566266841978, −13.65871104122632564860206170661, −13.16639203226429940120011308065, −12.81149062584364792423133025021, −11.72123289506996842259428660147, −11.35136314636757206169758619459, −10.82246833981666451061679104804, −10.30643596198444683353951289091, −9.173692047359594035107973941111, −8.619447060864028695969062358540, −8.404689753214811157991635364118, −7.32711469925858239422710865299, −6.20313830914802857444893198694, −6.08328547140778137441328859160, −4.92537639830739650212114525279, −3.77976963549535500715775465020, −2.95679205756170346011661390547, −1.16406634444353320136248559263,
1.16406634444353320136248559263, 2.95679205756170346011661390547, 3.77976963549535500715775465020, 4.92537639830739650212114525279, 6.08328547140778137441328859160, 6.20313830914802857444893198694, 7.32711469925858239422710865299, 8.404689753214811157991635364118, 8.619447060864028695969062358540, 9.173692047359594035107973941111, 10.30643596198444683353951289091, 10.82246833981666451061679104804, 11.35136314636757206169758619459, 11.72123289506996842259428660147, 12.81149062584364792423133025021, 13.16639203226429940120011308065, 13.65871104122632564860206170661, 14.48397440827174773566266841978, 14.64698277918031551293222793826, 15.57247524311043940173259685686
