| L(s) = 1 | − 4·4-s − 114·11-s + 16·16-s + 212·19-s + 420·29-s + 94·31-s − 12·41-s + 456·44-s − 155·49-s − 168·59-s + 112·61-s − 64·64-s + 720·71-s − 848·76-s + 320·79-s + 1.90e3·89-s − 726·101-s − 3.46e3·109-s − 1.68e3·116-s + 7.08e3·121-s − 376·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3.12·11-s + 1/4·16-s + 2.55·19-s + 2.68·29-s + 0.544·31-s − 0.0457·41-s + 1.56·44-s − 0.451·49-s − 0.370·59-s + 0.235·61-s − 1/8·64-s + 1.20·71-s − 1.27·76-s + 0.455·79-s + 2.27·89-s − 0.715·101-s − 3.04·109-s − 1.34·116-s + 5.32·121-s − 0.272·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.625134946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.625134946\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 155 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 57 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4642 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5942 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 210 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 47 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 101302 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 111490 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 17030 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291193 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 84 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 581362 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 565247 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 160 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 603349 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 954 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1788865 T^{2} + p^{6} T^{4} \) |
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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
−9.360476442195102426909377625070, −9.236573249697945980053961757728, −8.399949848473791044922229643162, −8.112005710953596558754537046565, −7.907735271204015252541322437778, −7.68446728093418935111293908007, −6.92356320768044654902762132567, −6.77064568934388662209729383080, −5.95021969662618112462855131914, −5.52818148011570479624502509953, −5.25525762468887170464745488654, −4.73930524845234838634584571955, −4.72152331123398583930310862577, −3.77535007477901229118472222940, −3.00328441577793788160038904869, −2.98386905272880015320848510951, −2.50199115792477931641273723951, −1.65801857273428066666728279876, −0.65990025893975089199837236689, −0.61479866302321414715485372095,
0.61479866302321414715485372095, 0.65990025893975089199837236689, 1.65801857273428066666728279876, 2.50199115792477931641273723951, 2.98386905272880015320848510951, 3.00328441577793788160038904869, 3.77535007477901229118472222940, 4.72152331123398583930310862577, 4.73930524845234838634584571955, 5.25525762468887170464745488654, 5.52818148011570479624502509953, 5.95021969662618112462855131914, 6.77064568934388662209729383080, 6.92356320768044654902762132567, 7.68446728093418935111293908007, 7.907735271204015252541322437778, 8.112005710953596558754537046565, 8.399949848473791044922229643162, 9.236573249697945980053961757728, 9.360476442195102426909377625070
