| L(s) = 1 | − 4·3-s − 2·5-s + 8·9-s − 4·11-s − 6·13-s + 8·15-s + 4·19-s + 2·25-s − 12·27-s + 6·29-s − 16·31-s + 16·33-s + 2·37-s + 24·39-s + 4·43-s − 16·45-s − 16·47-s − 2·49-s + 2·53-s + 8·55-s − 16·57-s − 12·59-s − 6·61-s + 12·65-s + 4·67-s − 8·75-s + 23·81-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.894·5-s + 8/3·9-s − 1.20·11-s − 1.66·13-s + 2.06·15-s + 0.917·19-s + 2/5·25-s − 2.30·27-s + 1.11·29-s − 2.87·31-s + 2.78·33-s + 0.328·37-s + 3.84·39-s + 0.609·43-s − 2.38·45-s − 2.33·47-s − 2/7·49-s + 0.274·53-s + 1.07·55-s − 2.11·57-s − 1.56·59-s − 0.768·61-s + 1.48·65-s + 0.488·67-s − 0.923·75-s + 23/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−10.75984207574315568072174627567, −10.58531895000847694922554123133, −9.747173572013814564060513999356, −9.708975134850384076379739323305, −9.062956555453450135335579972865, −8.162836717193120410632572548166, −7.79552032251226165920874563536, −7.46639080553063419008966883211, −6.88982122500765121758887663224, −6.65108369111800054916211922486, −5.67108331264863843287405349502, −5.61144621036201557340479689475, −5.05923233073200046327859911290, −4.74470352848300464199515850385, −4.14789643453972268610350955997, −3.28821402933475174139413419573, −2.60915342710743998365373176195, −1.47178569306772076606250035448, 0, 0,
1.47178569306772076606250035448, 2.60915342710743998365373176195, 3.28821402933475174139413419573, 4.14789643453972268610350955997, 4.74470352848300464199515850385, 5.05923233073200046327859911290, 5.61144621036201557340479689475, 5.67108331264863843287405349502, 6.65108369111800054916211922486, 6.88982122500765121758887663224, 7.46639080553063419008966883211, 7.79552032251226165920874563536, 8.162836717193120410632572548166, 9.062956555453450135335579972865, 9.708975134850384076379739323305, 9.747173572013814564060513999356, 10.58531895000847694922554123133, 10.75984207574315568072174627567
