| L(s) = 1 | + 4-s + 12·11-s + 16·19-s − 6·29-s + 8·31-s − 6·41-s + 12·44-s − 13·49-s − 12·59-s + 26·61-s − 64-s + 24·71-s + 16·76-s − 20·79-s + 36·89-s + 12·101-s + 28·109-s − 6·116-s + 58·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.61·11-s + 3.67·19-s − 1.11·29-s + 1.43·31-s − 0.937·41-s + 1.80·44-s − 1.85·49-s − 1.56·59-s + 3.32·61-s − 1/8·64-s + 2.84·71-s + 1.83·76-s − 2.25·79-s + 3.81·89-s + 1.19·101-s + 2.68·109-s − 0.557·116-s + 5.27·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.807299408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.807299408\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 35 T^{2} + 696 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 35 T^{2} - 3264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 85 T^{2} + 336 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.97193588989761362503316452261, −6.58721246122946685593871398548, −6.40956157443857190008851270885, −6.15685404959915365928806921668, −6.14337161470366828069746186692, −6.01834705771960148191738975580, −5.40100522328205667248069046929, −5.33468050562385146857043572243, −5.12847151162666513140143244543, −4.75886360451060189161550512102, −4.72877425985352543065887719277, −4.48140398582663885661055091326, −3.93076586080313024974586665576, −3.60285602823852269595081708751, −3.55116925225128066560049338097, −3.52700025763539797166124761016, −3.48904186335533102719937067109, −2.80517733237047729473536497888, −2.46767941547755616025477578686, −2.34394391002449516652780453906, −1.82085486482291096182027190950, −1.35725628447687316168321350547, −1.23765865277920913297382048458, −1.16043957127199207986154101104, −0.57219697366140985311191121789,
0.57219697366140985311191121789, 1.16043957127199207986154101104, 1.23765865277920913297382048458, 1.35725628447687316168321350547, 1.82085486482291096182027190950, 2.34394391002449516652780453906, 2.46767941547755616025477578686, 2.80517733237047729473536497888, 3.48904186335533102719937067109, 3.52700025763539797166124761016, 3.55116925225128066560049338097, 3.60285602823852269595081708751, 3.93076586080313024974586665576, 4.48140398582663885661055091326, 4.72877425985352543065887719277, 4.75886360451060189161550512102, 5.12847151162666513140143244543, 5.33468050562385146857043572243, 5.40100522328205667248069046929, 6.01834705771960148191738975580, 6.14337161470366828069746186692, 6.15685404959915365928806921668, 6.40956157443857190008851270885, 6.58721246122946685593871398548, 6.97193588989761362503316452261
