L(s) = 1 | + 2-s + 3·3-s − 2·4-s + 3·6-s − 6·7-s − 3·8-s + 2·9-s − 11-s − 6·12-s − 2·13-s − 6·14-s + 16-s − 6·17-s + 2·18-s − 18·21-s − 22-s − 13·23-s − 9·24-s − 2·26-s − 6·27-s + 12·28-s + 5·29-s + 11·31-s + 2·32-s − 3·33-s − 6·34-s − 4·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s − 4-s + 1.22·6-s − 2.26·7-s − 1.06·8-s + 2/3·9-s − 0.301·11-s − 1.73·12-s − 0.554·13-s − 1.60·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 3.92·21-s − 0.213·22-s − 2.71·23-s − 1.83·24-s − 0.392·26-s − 1.15·27-s + 2.26·28-s + 0.928·29-s + 1.97·31-s + 0.353·32-s − 0.522·33-s − 1.02·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.417769559\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.417769559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 181 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 273 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 21 T + 293 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
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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
−7.85798097625417361571397062123, −7.75752718063535588979639526869, −7.36280122613455449797895672376, −6.72708149568712833149822993123, −6.51044743068110102925065968786, −6.01098241864298337147488447676, −5.91928260601735850619997419039, −5.80303546096471051127776450172, −4.77679953874450840308947982058, −4.62016787427499746847381353621, −4.35448518424409690491723181240, −4.06161454083094428445882318802, −3.48524516652856102137724108632, −3.31261115520828932323504285019, −2.95184515086964323631421919604, −2.59139433048905446056033402104, −2.25829524104795198524923758294, −1.93878368770214206404028467889, −0.56791213225024985668349777347, −0.47203532683674249706226830995,
0.47203532683674249706226830995, 0.56791213225024985668349777347, 1.93878368770214206404028467889, 2.25829524104795198524923758294, 2.59139433048905446056033402104, 2.95184515086964323631421919604, 3.31261115520828932323504285019, 3.48524516652856102137724108632, 4.06161454083094428445882318802, 4.35448518424409690491723181240, 4.62016787427499746847381353621, 4.77679953874450840308947982058, 5.80303546096471051127776450172, 5.91928260601735850619997419039, 6.01098241864298337147488447676, 6.51044743068110102925065968786, 6.72708149568712833149822993123, 7.36280122613455449797895672376, 7.75752718063535588979639526869, 7.85798097625417361571397062123