| L(s) = 1 | − 3·3-s + 4·5-s + 2·7-s + 6·9-s − 5·11-s − 2·13-s − 12·15-s − 6·17-s − 2·19-s − 6·21-s + 6·23-s + 5·25-s − 9·27-s − 2·29-s + 4·31-s + 15·33-s + 8·35-s + 16·37-s + 6·39-s − 41-s − 7·43-s + 24·45-s − 2·47-s + 7·49-s + 18·51-s + 8·53-s − 20·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 0.755·7-s + 2·9-s − 1.50·11-s − 0.554·13-s − 3.09·15-s − 1.45·17-s − 0.458·19-s − 1.30·21-s + 1.25·23-s + 25-s − 1.73·27-s − 0.371·29-s + 0.718·31-s + 2.61·33-s + 1.35·35-s + 2.63·37-s + 0.960·39-s − 0.156·41-s − 1.06·43-s + 3.57·45-s − 0.291·47-s + 49-s + 2.52·51-s + 1.09·53-s − 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.199146135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199146135\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.82904414926888945141100165622, −10.54278886195656541682018265941, −10.29109408642528746595551147739, −9.589289455909210751780961107214, −9.583293321165901040547049164017, −8.847583932990132399624278974533, −8.263135505741028390028014532381, −7.76693174108882274756112681547, −7.20725774666102267819940473836, −6.54290021446911292507568138783, −6.51710117776102343224362163516, −5.67748646320448688626658378536, −5.52696458607118756525658402236, −5.10056379857187043365488284763, −4.55574452533685937713230057223, −4.25765311861324647021574533751, −2.82042901408217682863237725691, −2.31921188868815788905969804701, −1.75969376071989419692407391477, −0.69051615206741144692655257586,
0.69051615206741144692655257586, 1.75969376071989419692407391477, 2.31921188868815788905969804701, 2.82042901408217682863237725691, 4.25765311861324647021574533751, 4.55574452533685937713230057223, 5.10056379857187043365488284763, 5.52696458607118756525658402236, 5.67748646320448688626658378536, 6.51710117776102343224362163516, 6.54290021446911292507568138783, 7.20725774666102267819940473836, 7.76693174108882274756112681547, 8.263135505741028390028014532381, 8.847583932990132399624278974533, 9.583293321165901040547049164017, 9.589289455909210751780961107214, 10.29109408642528746595551147739, 10.54278886195656541682018265941, 10.82904414926888945141100165622
