| L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 3·9-s − 9·11-s + 2·13-s + 4·15-s − 17-s − 2·19-s + 2·21-s − 3·23-s + 3·25-s + 4·27-s − 6·29-s − 16·31-s − 18·33-s + 2·35-s − 11·37-s + 4·39-s + 11·41-s − 10·43-s + 6·45-s − 4·47-s − 9·49-s − 2·51-s − 53-s − 18·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 9-s − 2.71·11-s + 0.554·13-s + 1.03·15-s − 0.242·17-s − 0.458·19-s + 0.436·21-s − 0.625·23-s + 3/5·25-s + 0.769·27-s − 1.11·29-s − 2.87·31-s − 3.13·33-s + 0.338·35-s − 1.80·37-s + 0.640·39-s + 1.71·41-s − 1.52·43-s + 0.894·45-s − 0.583·47-s − 9/7·49-s − 0.280·51-s − 0.137·53-s − 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11 T + 108 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 120 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 134 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 126 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17 T + 228 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−7.924361543530100215286370325370, −7.66739975524794381391557172549, −7.29471335713089919147068281395, −6.96187486947307221570277406179, −6.44402167120497639031635131359, −6.10529746409550916311978646708, −5.49045682962790697419638842511, −5.48176086382127553624907816434, −4.93752262600133308022721566262, −4.87555161564514452426622178290, −3.96587065026384040944160029227, −3.90406283592132176560090088138, −3.15350069737746840164369875611, −3.09707261083776424017810187090, −2.42748496296284088419820241619, −2.13683565219453368005474543705, −1.70249550514622880985164625874, −1.48229324480088136082353184120, 0, 0,
1.48229324480088136082353184120, 1.70249550514622880985164625874, 2.13683565219453368005474543705, 2.42748496296284088419820241619, 3.09707261083776424017810187090, 3.15350069737746840164369875611, 3.90406283592132176560090088138, 3.96587065026384040944160029227, 4.87555161564514452426622178290, 4.93752262600133308022721566262, 5.48176086382127553624907816434, 5.49045682962790697419638842511, 6.10529746409550916311978646708, 6.44402167120497639031635131359, 6.96187486947307221570277406179, 7.29471335713089919147068281395, 7.66739975524794381391557172549, 7.924361543530100215286370325370
