| L(s) = 1 | − 2·5-s − 4·7-s + 2·11-s + 8·19-s − 8·23-s + 3·25-s + 4·29-s + 4·31-s + 8·35-s − 20·37-s + 4·41-s + 12·43-s − 8·47-s + 4·49-s − 4·53-s − 4·55-s − 4·59-s − 4·61-s + 8·67-s − 12·71-s − 16·73-s − 8·77-s + 16·79-s − 20·83-s − 12·89-s − 16·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 0.603·11-s + 1.83·19-s − 1.66·23-s + 3/5·25-s + 0.742·29-s + 0.718·31-s + 1.35·35-s − 3.28·37-s + 0.624·41-s + 1.82·43-s − 1.16·47-s + 4/7·49-s − 0.549·53-s − 0.539·55-s − 0.520·59-s − 0.512·61-s + 0.977·67-s − 1.42·71-s − 1.87·73-s − 0.911·77-s + 1.80·79-s − 2.19·83-s − 1.27·89-s − 1.64·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.245197891469318402408363034459, −7.943199732467613568871250431352, −7.34635731343169978895671569852, −7.33464826741388030590326826974, −6.70847398775694334947374868371, −6.62219431817092938687080852050, −5.98334254973009386340481409487, −5.87738189451312865979326026818, −5.20833229842077125247464652728, −4.98696135942463515839338435700, −4.21134064005229065019826663617, −4.10265662636460457490231649067, −3.56381535206122722128513294102, −3.26240289590185720383366273231, −2.86935265077217966263767276121, −2.45117204778861715199089250711, −1.43648368552214193129897200113, −1.24351988193469569641720271316, 0, 0,
1.24351988193469569641720271316, 1.43648368552214193129897200113, 2.45117204778861715199089250711, 2.86935265077217966263767276121, 3.26240289590185720383366273231, 3.56381535206122722128513294102, 4.10265662636460457490231649067, 4.21134064005229065019826663617, 4.98696135942463515839338435700, 5.20833229842077125247464652728, 5.87738189451312865979326026818, 5.98334254973009386340481409487, 6.62219431817092938687080852050, 6.70847398775694334947374868371, 7.33464826741388030590326826974, 7.34635731343169978895671569852, 7.943199732467613568871250431352, 8.245197891469318402408363034459
