| L(s) = 1 | − 3-s − 2·5-s + 7-s − 4·9-s − 11-s + 3·13-s + 2·15-s + 17-s + 3·19-s − 21-s + 2·23-s + 3·25-s + 6·27-s + 14·29-s + 7·31-s + 33-s − 2·35-s − 4·37-s − 3·39-s − 9·41-s + 8·45-s + 6·47-s − 12·49-s − 51-s + 8·53-s + 2·55-s − 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s − 4/3·9-s − 0.301·11-s + 0.832·13-s + 0.516·15-s + 0.242·17-s + 0.688·19-s − 0.218·21-s + 0.417·23-s + 3/5·25-s + 1.15·27-s + 2.59·29-s + 1.25·31-s + 0.174·33-s − 0.338·35-s − 0.657·37-s − 0.480·39-s − 1.40·41-s + 1.19·45-s + 0.875·47-s − 1.71·49-s − 0.140·51-s + 1.09·53-s + 0.269·55-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028978720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028978720\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 63 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 27 T + 375 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
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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
−8.086640794163801708984302080134, −7.87721004024048970374644774324, −7.39227872835667151359652745575, −6.97182845036459997513901874039, −6.68457135669706178736910489784, −6.24228378976600006290492488139, −6.12242537756754310281727521896, −5.57915394468678379516364858227, −5.12311705542106493069443961356, −5.07585733630567753845420230345, −4.47481851651348785179426817894, −4.38889528659987137940955615227, −3.53497137319422166219945193519, −3.38924588734296878222409657421, −2.97542692783780302677146515303, −2.67367340025602793065437752498, −2.05461850907676454989442022701, −1.36080929196599851574327985732, −0.77156776003339493902915315793, −0.52666957309215958415525374403,
0.52666957309215958415525374403, 0.77156776003339493902915315793, 1.36080929196599851574327985732, 2.05461850907676454989442022701, 2.67367340025602793065437752498, 2.97542692783780302677146515303, 3.38924588734296878222409657421, 3.53497137319422166219945193519, 4.38889528659987137940955615227, 4.47481851651348785179426817894, 5.07585733630567753845420230345, 5.12311705542106493069443961356, 5.57915394468678379516364858227, 6.12242537756754310281727521896, 6.24228378976600006290492488139, 6.68457135669706178736910489784, 6.97182845036459997513901874039, 7.39227872835667151359652745575, 7.87721004024048970374644774324, 8.086640794163801708984302080134
