| L(s) = 1 | + 4·5-s − 20·7-s + 60·11-s − 26·13-s + 84·17-s − 60·19-s + 72·23-s − 210·25-s + 124·29-s − 108·31-s − 80·35-s + 36·37-s + 52·41-s − 32·43-s + 428·47-s − 134·49-s − 380·53-s + 240·55-s + 1.42e3·59-s + 1.01e3·61-s − 104·65-s − 844·67-s + 868·71-s − 60·73-s − 1.20e3·77-s + 272·79-s + 1.25e3·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s − 1.07·7-s + 1.64·11-s − 0.554·13-s + 1.19·17-s − 0.724·19-s + 0.652·23-s − 1.67·25-s + 0.794·29-s − 0.625·31-s − 0.386·35-s + 0.159·37-s + 0.198·41-s − 0.113·43-s + 1.32·47-s − 0.390·49-s − 0.984·53-s + 0.588·55-s + 3.13·59-s + 2.12·61-s − 0.198·65-s − 1.53·67-s + 1.45·71-s − 0.0961·73-s − 1.77·77-s + 0.387·79-s + 1.65·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.226719849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.226719849\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 226 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 534 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 60 T + 3114 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 84 T + 10582 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 60 T + 6526 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 72 T + 14430 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 124 T - 1586 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 108 T - 16154 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 36 T + 42382 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 52 T + 76666 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 32 T + 150198 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 428 T + 116242 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 380 T + 258142 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1420 T + 882490 T^{2} - 1420 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1012 T + 708206 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 844 T + 778238 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 868 T + 888050 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 60 T + 114886 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 272 T + 993374 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1252 T + 1118698 T^{2} - 1252 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 572 T + 853306 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 708 T + 1762390 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.682769275316558451439543479409, −9.665501630742496776000765918800, −8.989548486388880517970525139888, −8.945050999339779608466859535843, −8.026538617283859789435278816216, −7.986602093722320726290349830754, −7.11642283516373095862941225541, −6.93627993051916138822679883669, −6.42675915290428761164413041458, −6.15425401009738511870859127689, −5.43878472926334727798313626109, −5.37583526258642100078448071333, −4.31517351953994171765573131581, −4.15661282717541198967742916102, −3.35076686402295415440951550779, −3.30871327573630775011803171414, −2.23935713191443860958381587503, −1.96978322746292201857488326676, −0.980845958631757754395181003044, −0.55044611406476825340072769790,
0.55044611406476825340072769790, 0.980845958631757754395181003044, 1.96978322746292201857488326676, 2.23935713191443860958381587503, 3.30871327573630775011803171414, 3.35076686402295415440951550779, 4.15661282717541198967742916102, 4.31517351953994171765573131581, 5.37583526258642100078448071333, 5.43878472926334727798313626109, 6.15425401009738511870859127689, 6.42675915290428761164413041458, 6.93627993051916138822679883669, 7.11642283516373095862941225541, 7.986602093722320726290349830754, 8.026538617283859789435278816216, 8.945050999339779608466859535843, 8.989548486388880517970525139888, 9.665501630742496776000765918800, 9.682769275316558451439543479409
