| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 4·10-s − 4·11-s − 6·12-s − 13-s − 4·14-s + 4·15-s + 5·16-s − 10·17-s + 6·18-s − 3·19-s − 6·20-s + 4·21-s − 8·22-s − 9·23-s − 8·24-s + 3·25-s − 2·26-s − 4·27-s − 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s − 1.20·11-s − 1.73·12-s − 0.277·13-s − 1.06·14-s + 1.03·15-s + 5/4·16-s − 2.42·17-s + 1.41·18-s − 0.688·19-s − 1.34·20-s + 0.872·21-s − 1.70·22-s − 1.87·23-s − 1.63·24-s + 3/5·25-s − 0.392·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 60 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 50 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 90 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 134 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 19 T + 252 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 140 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 246 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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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
−9.652420714269278564686753285487, −9.600516873136888235247570747972, −8.553488358181504791101811526665, −8.369103931345263489013047968798, −7.73589678570944847890062228996, −7.50335305222625085181300380662, −6.70137084399848694794174088545, −6.68809362990095016628647885460, −6.16905853612040079740785796708, −5.93895383136299238805627461161, −5.17864061466260350396003325547, −4.76951591052672606605992861028, −4.50612303761985982714189281352, −4.16743143517120259529648606362, −3.30027531686472465337447397895, −3.19765339415750741418118978281, −2.01926901614798684409134347927, −1.99602686798888578561163090851, 0, 0,
1.99602686798888578561163090851, 2.01926901614798684409134347927, 3.19765339415750741418118978281, 3.30027531686472465337447397895, 4.16743143517120259529648606362, 4.50612303761985982714189281352, 4.76951591052672606605992861028, 5.17864061466260350396003325547, 5.93895383136299238805627461161, 6.16905853612040079740785796708, 6.68809362990095016628647885460, 6.70137084399848694794174088545, 7.50335305222625085181300380662, 7.73589678570944847890062228996, 8.369103931345263489013047968798, 8.553488358181504791101811526665, 9.600516873136888235247570747972, 9.652420714269278564686753285487
