| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 3·9-s + 4·11-s − 6·12-s + 5·16-s + 8·17-s − 6·18-s + 8·19-s − 8·22-s + 2·23-s + 8·24-s − 8·25-s − 4·27-s + 4·29-s + 8·31-s − 6·32-s − 8·33-s − 16·34-s + 9·36-s + 4·37-s − 16·38-s + 8·43-s + 12·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 9-s + 1.20·11-s − 1.73·12-s + 5/4·16-s + 1.94·17-s − 1.41·18-s + 1.83·19-s − 1.70·22-s + 0.417·23-s + 1.63·24-s − 8/5·25-s − 0.769·27-s + 0.742·29-s + 1.43·31-s − 1.06·32-s − 1.39·33-s − 2.74·34-s + 3/2·36-s + 0.657·37-s − 2.59·38-s + 1.21·43-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.653949759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653949759\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 132 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 330 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−7.929810636848487045869753203908, −7.63851771410861133483413977239, −7.55953354123049374523907007887, −7.36305156012719539032890027922, −6.75897211754206122806484134861, −6.33809251644290952232583973150, −6.21234855577250091517407567289, −5.78773303949454301052930663970, −5.43373868011942525343699934744, −5.25191017736360935917729465429, −4.53260733618806271351060272761, −4.24334903567096370293880036266, −3.64843590593469576004920800334, −3.44646569434841982910688574208, −2.68223680626791196765246145153, −2.59550557341862992392642814660, −1.50162868975816338101711433979, −1.45041115334967069741516096869, −0.805607950369521101915310129232, −0.65815249507107071480907174263,
0.65815249507107071480907174263, 0.805607950369521101915310129232, 1.45041115334967069741516096869, 1.50162868975816338101711433979, 2.59550557341862992392642814660, 2.68223680626791196765246145153, 3.44646569434841982910688574208, 3.64843590593469576004920800334, 4.24334903567096370293880036266, 4.53260733618806271351060272761, 5.25191017736360935917729465429, 5.43373868011942525343699934744, 5.78773303949454301052930663970, 6.21234855577250091517407567289, 6.33809251644290952232583973150, 6.75897211754206122806484134861, 7.36305156012719539032890027922, 7.55953354123049374523907007887, 7.63851771410861133483413977239, 7.929810636848487045869753203908
