L(s) = 1 | − 2·3-s + 2·9-s − 2·11-s + 2·13-s + 4·17-s − 6·19-s − 6·27-s + 6·29-s + 16·31-s + 4·33-s − 6·37-s − 4·39-s + 10·43-s + 16·47-s + 10·49-s − 8·51-s + 10·53-s + 12·57-s + 6·59-s − 18·61-s − 10·67-s + 11·81-s − 2·83-s − 12·87-s − 32·93-s + 4·97-s − 4·99-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 2/3·9-s − 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s − 1.15·27-s + 1.11·29-s + 2.87·31-s + 0.696·33-s − 0.986·37-s − 0.640·39-s + 1.52·43-s + 2.33·47-s + 10/7·49-s − 1.12·51-s + 1.37·53-s + 1.58·57-s + 0.781·59-s − 2.30·61-s − 1.22·67-s + 11/9·81-s − 0.219·83-s − 1.28·87-s − 3.31·93-s + 0.406·97-s − 0.402·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.480508866\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.480508866\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.872932570322977685621864201960, −9.055087284914310541743538158781, −8.741725797543948219256329406089, −8.599073138777116856074738628014, −7.895540715125077742601543027210, −7.56250096929534773600998650581, −7.28393738337688952010220885128, −6.63568898522911555025553376384, −6.20228723236467261597295882552, −6.03780032195121777799909554856, −5.56488020806183165662227679280, −5.23603551650315258810324566151, −4.47398920521395097767225012078, −4.36555707860134889009954967928, −3.85977087013884148245362238782, −3.05563059111024567756532877868, −2.61337118097625187623038090185, −2.03026736438775170760413857794, −1.05445676113746192266653822891, −0.62372294225937580169693360239,
0.62372294225937580169693360239, 1.05445676113746192266653822891, 2.03026736438775170760413857794, 2.61337118097625187623038090185, 3.05563059111024567756532877868, 3.85977087013884148245362238782, 4.36555707860134889009954967928, 4.47398920521395097767225012078, 5.23603551650315258810324566151, 5.56488020806183165662227679280, 6.03780032195121777799909554856, 6.20228723236467261597295882552, 6.63568898522911555025553376384, 7.28393738337688952010220885128, 7.56250096929534773600998650581, 7.895540715125077742601543027210, 8.599073138777116856074738628014, 8.741725797543948219256329406089, 9.055087284914310541743538158781, 9.872932570322977685621864201960