| L(s) = 1 | + 6·3-s + 21·9-s − 18·11-s + 12·19-s + 2·25-s + 54·27-s − 108·33-s + 18·41-s + 18·43-s − 14·49-s + 72·57-s − 18·59-s − 6·67-s + 20·73-s + 12·75-s + 108·81-s + 36·83-s + 2·97-s − 378·99-s + 48·113-s + 167·121-s + 108·123-s + 127-s + 108·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 7·9-s − 5.42·11-s + 2.75·19-s + 2/5·25-s + 10.3·27-s − 18.8·33-s + 2.81·41-s + 2.74·43-s − 2·49-s + 9.53·57-s − 2.34·59-s − 0.733·67-s + 2.34·73-s + 1.38·75-s + 12·81-s + 3.95·83-s + 0.203·97-s − 37.9·99-s + 4.51·113-s + 15.1·121-s + 9.73·123-s + 0.0887·127-s + 9.50·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.50861638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.50861638\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 2 T^{2} - 165 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 26 T^{2} + 147 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 26 T^{2} - 165 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 22 T^{2} - 1725 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^3$ | \( 1 + 26 T^{2} - 3045 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 - 18 T + 191 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.45174848427704160086987273644, −7.04772016776026988750794974514, −6.90928699014468036899918917940, −6.16599244793007041438694527001, −6.07805891935055811266064797121, −5.96600283097670784252017434950, −5.64198486780844421031088939870, −5.21152481912439668863685235386, −5.08783234825062684482131050671, −5.01526970773282503402051061355, −4.59146886092988384836453472701, −4.55624258852374350441030299264, −4.30151452355648023469411796665, −3.57444050890298993194600492889, −3.49864711531740078752660333351, −3.45755448290330726385707385480, −2.86997155378123363257134970115, −2.86694506125260715624793781055, −2.85673318581911516644869673276, −2.41018230680628188104764811609, −2.24667719814692561665018610993, −2.02347074874090526027698416493, −1.61543965145948125269807704952, −0.77799970049017416108261013745, −0.68582904099563324611671374407,
0.68582904099563324611671374407, 0.77799970049017416108261013745, 1.61543965145948125269807704952, 2.02347074874090526027698416493, 2.24667719814692561665018610993, 2.41018230680628188104764811609, 2.85673318581911516644869673276, 2.86694506125260715624793781055, 2.86997155378123363257134970115, 3.45755448290330726385707385480, 3.49864711531740078752660333351, 3.57444050890298993194600492889, 4.30151452355648023469411796665, 4.55624258852374350441030299264, 4.59146886092988384836453472701, 5.01526970773282503402051061355, 5.08783234825062684482131050671, 5.21152481912439668863685235386, 5.64198486780844421031088939870, 5.96600283097670784252017434950, 6.07805891935055811266064797121, 6.16599244793007041438694527001, 6.90928699014468036899918917940, 7.04772016776026988750794974514, 7.45174848427704160086987273644
