| L(s) = 1 | − 2·3-s − 3·5-s + 2·7-s + 3·9-s − 2·11-s − 5·13-s + 6·15-s + 2·17-s + 9·19-s − 4·21-s − 2·23-s + 5·25-s − 4·27-s − 3·29-s − 16·31-s + 4·33-s − 6·35-s − 5·37-s + 10·39-s − 10·41-s + 10·43-s − 9·45-s + 47-s + 3·49-s − 4·51-s − 20·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 0.755·7-s + 9-s − 0.603·11-s − 1.38·13-s + 1.54·15-s + 0.485·17-s + 2.06·19-s − 0.872·21-s − 0.417·23-s + 25-s − 0.769·27-s − 0.557·29-s − 2.87·31-s + 0.696·33-s − 1.01·35-s − 0.821·37-s + 1.60·39-s − 1.56·41-s + 1.52·43-s − 1.34·45-s + 0.145·47-s + 3/7·49-s − 0.560·51-s − 2.74·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 72 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 86 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 122 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 66 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.142866583232895902701417482665, −8.686256437389933871526925666902, −7.77735130341480161855656883339, −7.76316581981646369724033447260, −7.56178852228549957839936062626, −7.40225518436017348266008794347, −6.63833068251119298999507673294, −6.36266198335412773342705106865, −5.51985962984313655979795981122, −5.34637780706155344972664767845, −4.96958999771083961580095206452, −4.85727617426182684057928997651, −3.98786619463098239908850848582, −3.72884068068349385326566744734, −3.21635951752728731960723666597, −2.59446973183155979355063936592, −1.70840668097776916950195189942, −1.29732679535621725178380720317, 0, 0,
1.29732679535621725178380720317, 1.70840668097776916950195189942, 2.59446973183155979355063936592, 3.21635951752728731960723666597, 3.72884068068349385326566744734, 3.98786619463098239908850848582, 4.85727617426182684057928997651, 4.96958999771083961580095206452, 5.34637780706155344972664767845, 5.51985962984313655979795981122, 6.36266198335412773342705106865, 6.63833068251119298999507673294, 7.40225518436017348266008794347, 7.56178852228549957839936062626, 7.76316581981646369724033447260, 7.77735130341480161855656883339, 8.686256437389933871526925666902, 9.142866583232895902701417482665
