| L(s) = 1 | − 2·3-s + 3·9-s − 2·11-s + 2·19-s − 4·27-s + 4·33-s + 2·43-s + 2·49-s − 4·57-s + 2·59-s + 2·67-s + 5·81-s − 2·83-s + 4·89-s − 6·99-s + 2·107-s + 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·3-s + 3·9-s − 2·11-s + 2·19-s − 4·27-s + 4·33-s + 2·43-s + 2·49-s − 4·57-s + 2·59-s + 2·67-s + 5·81-s − 2·83-s + 4·89-s − 6·99-s + 2·107-s + 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6814591661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6814591661\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + T^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.134789036095722181706723046940, −8.803168269050532159351772513753, −8.126544118231910473478541616356, −7.69989132846954318099785318847, −7.47583931653308452568269025400, −7.32920848998475293440584309054, −6.72810762523987802388296506861, −6.47537485738457200164339407308, −5.73292479242823631303042864156, −5.56503528388875144518949146147, −5.45892221723750339750860124333, −5.05058586032667330641143338617, −4.52430033146478063012974348020, −4.22964232404430460503467035321, −3.57151940899356055238328739902, −3.17825728774416267984895776629, −2.25057310954275461927680389280, −2.21222672955083138994476016064, −0.988624387561853365470562512342, −0.75825816710647015445463612734,
0.75825816710647015445463612734, 0.988624387561853365470562512342, 2.21222672955083138994476016064, 2.25057310954275461927680389280, 3.17825728774416267984895776629, 3.57151940899356055238328739902, 4.22964232404430460503467035321, 4.52430033146478063012974348020, 5.05058586032667330641143338617, 5.45892221723750339750860124333, 5.56503528388875144518949146147, 5.73292479242823631303042864156, 6.47537485738457200164339407308, 6.72810762523987802388296506861, 7.32920848998475293440584309054, 7.47583931653308452568269025400, 7.69989132846954318099785318847, 8.126544118231910473478541616356, 8.803168269050532159351772513753, 9.134789036095722181706723046940
