| L(s) = 1 | + 4·5-s + 52·9-s − 80·11-s + 80·19-s + 146·25-s − 280·29-s + 384·31-s − 1.04e3·41-s + 208·45-s + 932·49-s − 320·55-s + 1.39e3·59-s + 1.38e3·61-s + 1.37e3·71-s + 1.47e3·79-s + 954·81-s + 1.32e3·89-s + 320·95-s − 4.16e3·99-s + 1.41e3·101-s − 216·109-s + 1.74e3·121-s + 1.60e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 1.92·9-s − 2.19·11-s + 0.965·19-s + 1.16·25-s − 1.79·29-s + 2.22·31-s − 3.99·41-s + 0.689·45-s + 2.71·49-s − 0.784·55-s + 3.07·59-s + 2.90·61-s + 2.30·71-s + 2.09·79-s + 1.30·81-s + 1.57·89-s + 0.345·95-s − 4.22·99-s + 1.39·101-s − 0.189·109-s + 1.31·121-s + 1.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.852499412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.852499412\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T - 26 p T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1750 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 932 T^{2} + 448998 T^{4} - 932 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 40 T + 1526 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3188 T^{2} + 11156118 T^{4} - 3188 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2756 T^{2} - 6449082 T^{4} - 2756 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 40 T + 12582 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 228 T^{2} - 31055962 T^{4} - 228 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 140 T + 47534 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 192 T + 13502 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 127124 T^{2} + 8061820662 T^{4} - 127124 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 524 T + 203030 T^{2} + 524 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 227668 T^{2} + 24136249398 T^{4} - 227668 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 208964 T^{2} + 31759440582 T^{4} - 208964 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 29100 T^{2} + 19096738358 T^{4} + 29100 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 696 T + 346006 T^{2} - 696 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 692 T + 554862 T^{2} - 692 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1150772 T^{2} + 511868382678 T^{4} - 1150772 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 688 T + 809582 T^{2} - 688 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 934628 T^{2} + 433634093478 T^{4} - 934628 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 736 T + 1115358 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 846708 T^{2} + 680383628438 T^{4} - 846708 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 660 T + 1463542 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 3172484 T^{2} + 4151279558022 T^{4} - 3172484 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.974389790177209400240482610961, −7.73694661668826087774957016489, −7.48949394496358305914742567294, −7.18097755717337155894013348753, −7.03527839650686385237104183183, −6.62621242036411140440263229096, −6.62406624286703610929079267306, −6.42455030419438343669042003662, −5.63160974561454941839569233864, −5.52386615201605147612056556416, −5.29172547399707346140844087125, −5.11966644225708985223283247360, −4.93683730693395797513667084186, −4.56494430609073835159457240524, −4.03510045783882149761245339080, −3.94047977113667131324845663061, −3.49562807113466535314530068425, −3.25988905655850823428039751436, −2.81209772901212244713339272471, −2.31128292180021585852050914169, −2.02489550531231270614042593216, −1.99446929662126804942197754958, −1.02499901939762239527508913888, −0.886532323813446540171008027541, −0.46469174309031608193399381287,
0.46469174309031608193399381287, 0.886532323813446540171008027541, 1.02499901939762239527508913888, 1.99446929662126804942197754958, 2.02489550531231270614042593216, 2.31128292180021585852050914169, 2.81209772901212244713339272471, 3.25988905655850823428039751436, 3.49562807113466535314530068425, 3.94047977113667131324845663061, 4.03510045783882149761245339080, 4.56494430609073835159457240524, 4.93683730693395797513667084186, 5.11966644225708985223283247360, 5.29172547399707346140844087125, 5.52386615201605147612056556416, 5.63160974561454941839569233864, 6.42455030419438343669042003662, 6.62406624286703610929079267306, 6.62621242036411140440263229096, 7.03527839650686385237104183183, 7.18097755717337155894013348753, 7.48949394496358305914742567294, 7.73694661668826087774957016489, 7.974389790177209400240482610961
