| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 7-s − 4·8-s + 2·9-s + 4·10-s − 6·11-s + 6·12-s + 2·13-s + 2·14-s − 4·15-s + 5·16-s + 7·17-s − 4·18-s − 3·19-s − 6·20-s − 2·21-s + 12·22-s − 5·23-s − 8·24-s + 3·25-s − 4·26-s + 6·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 0.377·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.80·11-s + 1.73·12-s + 0.554·13-s + 0.534·14-s − 1.03·15-s + 5/4·16-s + 1.69·17-s − 0.942·18-s − 0.688·19-s − 1.34·20-s − 0.436·21-s + 2.55·22-s − 1.04·23-s − 1.63·24-s + 3/5·25-s − 0.784·26-s + 1.15·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16240900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16240900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.726028963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726028963\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 41 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 17 T + 129 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 63 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 62 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 119 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 282 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 157 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 223 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 303 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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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
−8.366938359377628494836474412651, −8.179341830867882837611224862542, −8.041697403633572916401856560805, −7.977646860714968706792712680464, −7.33558156980303238954473259120, −6.98039841695498363017039452487, −6.60784931159788051487961453598, −6.35764338974755101730257871238, −5.59461382075425144420489285261, −5.43922343435350674725800873453, −4.89561552692519194667259149296, −4.30025659602895525412134912434, −3.86890472346090310527243860383, −3.42297614653404103395128370945, −2.89244802660586351583907479712, −2.79532279395716812111339148804, −2.30136046449261034835897949647, −1.71055631895506360067940748217, −0.807348533499459926017870989737, −0.61849767030666140764465821385,
0.61849767030666140764465821385, 0.807348533499459926017870989737, 1.71055631895506360067940748217, 2.30136046449261034835897949647, 2.79532279395716812111339148804, 2.89244802660586351583907479712, 3.42297614653404103395128370945, 3.86890472346090310527243860383, 4.30025659602895525412134912434, 4.89561552692519194667259149296, 5.43922343435350674725800873453, 5.59461382075425144420489285261, 6.35764338974755101730257871238, 6.60784931159788051487961453598, 6.98039841695498363017039452487, 7.33558156980303238954473259120, 7.977646860714968706792712680464, 8.041697403633572916401856560805, 8.179341830867882837611224862542, 8.366938359377628494836474412651
