L(s) = 1 | + 3·3-s + 7-s + 2·9-s − 2·11-s − 8·13-s − 17-s + 3·21-s + 3·23-s − 6·27-s − 5·29-s + 6·31-s − 6·33-s − 16·37-s − 24·39-s − 6·41-s − 12·43-s + 6·47-s − 2·49-s − 3·51-s − 3·53-s − 10·59-s − 11·61-s + 2·63-s + 16·67-s + 9·69-s + 6·71-s − 23·73-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2/3·9-s − 0.603·11-s − 2.21·13-s − 0.242·17-s + 0.654·21-s + 0.625·23-s − 1.15·27-s − 0.928·29-s + 1.07·31-s − 1.04·33-s − 2.63·37-s − 3.84·39-s − 0.937·41-s − 1.82·43-s + 0.875·47-s − 2/7·49-s − 0.420·51-s − 0.412·53-s − 1.30·59-s − 1.40·61-s + 0.251·63-s + 1.95·67-s + 1.08·69-s + 0.712·71-s − 2.69·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 133 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 123 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 23 T + 277 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 27 T + 337 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 193 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.259741115214459591493335730899, −7.968275165552492968490092800514, −7.35428668053940891074640686375, −7.34332128446837249426664486604, −6.90522295290704701884676870667, −6.62635956897337541844045736723, −5.74835161620224278970938375701, −5.63953997479103078442159367514, −4.96333895446597968853112499311, −4.95004495714196614097970491129, −4.41632042810119722171234999991, −3.93777399207267259103718415756, −3.25357155132373839925404481607, −3.11576859801955217031266819624, −2.73017203621577420589298247557, −2.37413644325933034098614993449, −1.72256025701593594240909040968, −1.58162896782745415033246615935, 0, 0,
1.58162896782745415033246615935, 1.72256025701593594240909040968, 2.37413644325933034098614993449, 2.73017203621577420589298247557, 3.11576859801955217031266819624, 3.25357155132373839925404481607, 3.93777399207267259103718415756, 4.41632042810119722171234999991, 4.95004495714196614097970491129, 4.96333895446597968853112499311, 5.63953997479103078442159367514, 5.74835161620224278970938375701, 6.62635956897337541844045736723, 6.90522295290704701884676870667, 7.34332128446837249426664486604, 7.35428668053940891074640686375, 7.968275165552492968490092800514, 8.259741115214459591493335730899