| L(s) = 1 | + 3·3-s + 3·5-s + 4·9-s + 2·11-s + 6·13-s + 9·15-s + 2·17-s + 12·19-s + 2·23-s + 6·27-s − 10·29-s + 17·31-s + 6·33-s + 18·39-s − 41-s − 5·43-s + 12·45-s − 20·47-s + 6·51-s + 5·53-s + 6·55-s + 36·57-s + 4·59-s + 5·61-s + 18·65-s + 9·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 4/3·9-s + 0.603·11-s + 1.66·13-s + 2.32·15-s + 0.485·17-s + 2.75·19-s + 0.417·23-s + 1.15·27-s − 1.85·29-s + 3.05·31-s + 1.04·33-s + 2.88·39-s − 0.156·41-s − 0.762·43-s + 1.78·45-s − 2.91·47-s + 0.840·51-s + 0.686·53-s + 0.809·55-s + 4.76·57-s + 0.520·59-s + 0.640·61-s + 2.23·65-s + 1.09·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.84156361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.84156361\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_4$ | \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 17 T + 131 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_4$ | \( 1 + 5 T + 89 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 109 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 125 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 85 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 221 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.682387309683576626356959481357, −8.506688500161231004311711585035, −8.108448796978046193475231524049, −7.961124940640164801546031569083, −7.43376244387374215714960849448, −6.83359675553433149076650620813, −6.73764853677029265620042500565, −6.16257719436519209809635571122, −5.84620362911401061994813275465, −5.42672818410414175540176096598, −5.04154532832702039834195383205, −4.60398173680048519162805186593, −3.77203565043264282322181540255, −3.66158674754342727022811804463, −3.10613542763688161550250061349, −2.99898620483043452236770836242, −2.32490400693043135370915291419, −1.82186369562600974388642463745, −1.17417811731172735545052185636, −1.11457962680886830333445368701,
1.11457962680886830333445368701, 1.17417811731172735545052185636, 1.82186369562600974388642463745, 2.32490400693043135370915291419, 2.99898620483043452236770836242, 3.10613542763688161550250061349, 3.66158674754342727022811804463, 3.77203565043264282322181540255, 4.60398173680048519162805186593, 5.04154532832702039834195383205, 5.42672818410414175540176096598, 5.84620362911401061994813275465, 6.16257719436519209809635571122, 6.73764853677029265620042500565, 6.83359675553433149076650620813, 7.43376244387374215714960849448, 7.961124940640164801546031569083, 8.108448796978046193475231524049, 8.506688500161231004311711585035, 8.682387309683576626356959481357
