L(s) = 1 | + 5-s + 3·7-s − 2·11-s − 5·13-s − 5·17-s − 7·19-s + 5·23-s − 25-s + 3·29-s + 7·31-s + 3·35-s − 6·37-s − 12·41-s − 8·43-s + 3·47-s + 49-s + 10·53-s − 2·55-s − 14·59-s + 61-s − 5·65-s − 4·67-s − 8·71-s − 7·73-s − 6·77-s − 7·79-s − 25·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.603·11-s − 1.38·13-s − 1.21·17-s − 1.60·19-s + 1.04·23-s − 1/5·25-s + 0.557·29-s + 1.25·31-s + 0.507·35-s − 0.986·37-s − 1.87·41-s − 1.21·43-s + 0.437·47-s + 1/7·49-s + 1.37·53-s − 0.269·55-s − 1.82·59-s + 0.128·61-s − 0.620·65-s − 0.488·67-s − 0.949·71-s − 0.819·73-s − 0.683·77-s − 0.787·79-s − 2.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
good | 5 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 44 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 88 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 105 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 162 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 25 T + 314 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.935561999350700627232756828318, −7.84009843017776050510162243778, −7.27885227674923987723281771784, −6.86268118760155579687912630363, −6.57157191076270476884801171760, −6.54061077805822248954601543398, −5.61039047914606034098155311323, −5.52424190293853526885104202332, −5.09720248416829243112674753578, −4.60924315021771887728562651724, −4.44410459180859756168035091557, −4.26733118436619383989946041498, −3.36985646069599456312270038851, −2.96477905847859692682043104667, −2.42944332300122310886360353692, −2.30074328053078635077136756871, −1.55108157117675645149728519860, −1.38301966495389704652907554820, 0, 0,
1.38301966495389704652907554820, 1.55108157117675645149728519860, 2.30074328053078635077136756871, 2.42944332300122310886360353692, 2.96477905847859692682043104667, 3.36985646069599456312270038851, 4.26733118436619383989946041498, 4.44410459180859756168035091557, 4.60924315021771887728562651724, 5.09720248416829243112674753578, 5.52424190293853526885104202332, 5.61039047914606034098155311323, 6.54061077805822248954601543398, 6.57157191076270476884801171760, 6.86268118760155579687912630363, 7.27885227674923987723281771784, 7.84009843017776050510162243778, 7.935561999350700627232756828318