| L(s) = 1 | − 3·4-s + 9-s + 4·16-s + 2·19-s + 32·29-s + 16·31-s − 3·36-s + 13·49-s + 20·59-s − 14·61-s − 9·64-s − 48·71-s − 6·76-s − 14·79-s − 12·89-s + 32·101-s − 30·109-s − 96·116-s + 22·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1/3·9-s + 16-s + 0.458·19-s + 5.94·29-s + 2.87·31-s − 1/2·36-s + 13/7·49-s + 2.60·59-s − 1.79·61-s − 9/8·64-s − 5.69·71-s − 0.688·76-s − 1.57·79-s − 1.27·89-s + 3.18·101-s − 2.87·109-s − 8.91·116-s + 2·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.006665974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.006665974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 6 T^{2} - 2173 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 25 T^{2} - 4704 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.147753213175085750875338697775, −7.65165688483612402700419137938, −7.20091891996140394553794202949, −7.19867953340156791415853024226, −7.03653618130354693691204036329, −6.42751053576422458762043167726, −6.31231722360307485970168648912, −6.22036747044185297132608661203, −6.09231289050230307137136399550, −5.43073996210336819989443082228, −5.30335368193186599788513691409, −5.06632342834655948521222583486, −4.71293603732825749710711361270, −4.41957139837792523217893035856, −4.28671237161099687022855409231, −4.27367659685432180208384723539, −4.00581939954847360424214412817, −3.13226423094054427681535094403, −2.86078301210845629961313754186, −2.85955705465982351805212457988, −2.82094220221747914281718755376, −1.95649035390056783314082261717, −1.25914278255035394162404658719, −1.04873917174459805634017296509, −0.61682798557765087417713721348,
0.61682798557765087417713721348, 1.04873917174459805634017296509, 1.25914278255035394162404658719, 1.95649035390056783314082261717, 2.82094220221747914281718755376, 2.85955705465982351805212457988, 2.86078301210845629961313754186, 3.13226423094054427681535094403, 4.00581939954847360424214412817, 4.27367659685432180208384723539, 4.28671237161099687022855409231, 4.41957139837792523217893035856, 4.71293603732825749710711361270, 5.06632342834655948521222583486, 5.30335368193186599788513691409, 5.43073996210336819989443082228, 6.09231289050230307137136399550, 6.22036747044185297132608661203, 6.31231722360307485970168648912, 6.42751053576422458762043167726, 7.03653618130354693691204036329, 7.19867953340156791415853024226, 7.20091891996140394553794202949, 7.65165688483612402700419137938, 8.147753213175085750875338697775
