| L(s) = 1 | − 2·7-s − 4·13-s − 2·19-s − 8·23-s + 2·25-s + 8·29-s + 8·31-s − 4·37-s − 12·41-s − 12·47-s + 3·49-s − 8·59-s + 12·61-s + 16·67-s − 20·71-s + 12·73-s + 4·83-s + 4·89-s + 8·91-s + 20·97-s + 16·103-s + 4·107-s − 12·109-s − 16·113-s − 22·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.10·13-s − 0.458·19-s − 1.66·23-s + 2/5·25-s + 1.48·29-s + 1.43·31-s − 0.657·37-s − 1.87·41-s − 1.75·47-s + 3/7·49-s − 1.04·59-s + 1.53·61-s + 1.95·67-s − 2.37·71-s + 1.40·73-s + 0.439·83-s + 0.423·89-s + 0.838·91-s + 2.03·97-s + 1.57·103-s + 0.386·107-s − 1.14·109-s − 1.50·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 230 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.49319518637628585033979555324, −7.12280852183328689774986411995, −6.73317494989573537807050785996, −6.52138914949929911361594868312, −6.14330914560680280850677665193, −6.12786390766243469516063658362, −5.26134955269787280389645544576, −5.10645268147226859492584833635, −4.79195791719981943224983072364, −4.51764354524026810845626081871, −3.79119547248927746536186298081, −3.75900609362930158354664743603, −3.21796959530136593973094686599, −2.80952444463012360969730288573, −2.31929969940564237489543718826, −2.18169991225806667497019263958, −1.44140656629989608823028780018, −0.967189917001944612391575457290, 0, 0,
0.967189917001944612391575457290, 1.44140656629989608823028780018, 2.18169991225806667497019263958, 2.31929969940564237489543718826, 2.80952444463012360969730288573, 3.21796959530136593973094686599, 3.75900609362930158354664743603, 3.79119547248927746536186298081, 4.51764354524026810845626081871, 4.79195791719981943224983072364, 5.10645268147226859492584833635, 5.26134955269787280389645544576, 6.12786390766243469516063658362, 6.14330914560680280850677665193, 6.52138914949929911361594868312, 6.73317494989573537807050785996, 7.12280852183328689774986411995, 7.49319518637628585033979555324
