| L(s) = 1 | + 2·3-s − 9-s − 2·11-s + 2·13-s + 2·17-s − 4·19-s − 4·23-s − 6·27-s + 2·29-s − 12·31-s − 4·33-s − 4·37-s + 4·39-s − 12·41-s + 12·43-s + 18·47-s + 4·51-s − 16·53-s − 8·57-s − 12·59-s + 8·61-s − 8·69-s + 12·71-s − 16·73-s − 18·79-s − 4·81-s + 4·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 1.15·27-s + 0.371·29-s − 2.15·31-s − 0.696·33-s − 0.657·37-s + 0.640·39-s − 1.87·41-s + 1.82·43-s + 2.62·47-s + 0.560·51-s − 2.19·53-s − 1.05·57-s − 1.56·59-s + 1.02·61-s − 0.963·69-s + 1.42·71-s − 1.87·73-s − 2.02·79-s − 4/9·81-s + 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 173 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 132 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 207 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 193 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.52113495348906143262589289947, −7.39676500986957984499470522260, −6.91646346724984312387936557820, −6.42272264596227450637708848581, −6.09406582688223523850838092827, −5.87517400212157927668328972941, −5.35093948169453819984560555352, −5.28990264258062854828645911310, −4.66283014116644431737492337169, −4.25413191270887747525442371826, −3.76242371248974508147345321937, −3.72594029424690463082110300523, −3.01787677988875218152710047086, −3.00579399525752819305038683236, −2.25297330890949067344396865483, −2.21995623750947290692099850188, −1.58467906614840451463958115281, −1.12582916080466871028249856534, 0, 0,
1.12582916080466871028249856534, 1.58467906614840451463958115281, 2.21995623750947290692099850188, 2.25297330890949067344396865483, 3.00579399525752819305038683236, 3.01787677988875218152710047086, 3.72594029424690463082110300523, 3.76242371248974508147345321937, 4.25413191270887747525442371826, 4.66283014116644431737492337169, 5.28990264258062854828645911310, 5.35093948169453819984560555352, 5.87517400212157927668328972941, 6.09406582688223523850838092827, 6.42272264596227450637708848581, 6.91646346724984312387936557820, 7.39676500986957984499470522260, 7.52113495348906143262589289947
