| L(s) = 1 | + 2-s + 4-s + 8-s − 6·11-s − 5·13-s + 16-s − 6·22-s − 3·23-s + 8·25-s − 5·26-s + 32-s + 37-s − 6·44-s − 3·46-s − 12·47-s − 10·49-s + 8·50-s − 5·52-s − 3·59-s − 11·61-s + 64-s + 3·71-s + 4·73-s + 74-s − 12·83-s − 6·88-s − 3·92-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s − 1.38·13-s + 1/4·16-s − 1.27·22-s − 0.625·23-s + 8/5·25-s − 0.980·26-s + 0.176·32-s + 0.164·37-s − 0.904·44-s − 0.442·46-s − 1.75·47-s − 1.42·49-s + 1.13·50-s − 0.693·52-s − 0.390·59-s − 1.40·61-s + 1/8·64-s + 0.356·71-s + 0.468·73-s + 0.116·74-s − 1.31·83-s − 0.639·88-s − 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−8.670274077605398649960751251650, −8.240747879232697714406583035573, −7.77529422590753632701004175077, −7.42748451339827369800413996854, −6.88149477971273576832520832473, −6.37127546589436624514450600258, −5.80282719471908383140327704296, −5.04999835347998172852912181029, −4.98480653433086448867763565630, −4.48214866038882183870748168335, −3.55024271220538671458349359859, −2.84037249440615096555350895894, −2.59658973066534259191127744193, −1.64914251233014776706705464060, 0,
1.64914251233014776706705464060, 2.59658973066534259191127744193, 2.84037249440615096555350895894, 3.55024271220538671458349359859, 4.48214866038882183870748168335, 4.98480653433086448867763565630, 5.04999835347998172852912181029, 5.80282719471908383140327704296, 6.37127546589436624514450600258, 6.88149477971273576832520832473, 7.42748451339827369800413996854, 7.77529422590753632701004175077, 8.240747879232697714406583035573, 8.670274077605398649960751251650
