| L(s) = 1 | + 2-s + 4-s + 8-s − 3·11-s − 2·13-s + 16-s − 3·22-s + 12·23-s − 7·25-s − 2·26-s + 32-s + 10·37-s − 3·44-s + 12·46-s + 6·47-s + 5·49-s − 7·50-s − 2·52-s + 12·59-s + 4·61-s + 64-s + 73-s + 10·74-s + 6·83-s − 3·88-s + 12·92-s + 6·94-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.904·11-s − 0.554·13-s + 1/4·16-s − 0.639·22-s + 2.50·23-s − 7/5·25-s − 0.392·26-s + 0.176·32-s + 1.64·37-s − 0.452·44-s + 1.76·46-s + 0.875·47-s + 5/7·49-s − 0.989·50-s − 0.277·52-s + 1.56·59-s + 0.512·61-s + 1/8·64-s + 0.117·73-s + 1.16·74-s + 0.658·83-s − 0.319·88-s + 1.25·92-s + 0.618·94-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
| \(L(1)\) |
\(\approx\) |
\(2.517063610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.517063610\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.026129019908772187071485135707, −8.613939219671200765460315094925, −7.933617314069736134368829467402, −7.55756739866551106279076578094, −7.16821278301953870536747588848, −6.66802986371476306529165742036, −6.02136980541090391017209854900, −5.50325664724285495618660757850, −5.13364610600543206575722297600, −4.57730489606276588333642305309, −4.00517135852989304461611093037, −3.30078339073299021127094761027, −2.64139272167887492536232147129, −2.19862661617361079661334145404, −0.912022968073060594850037855773,
0.912022968073060594850037855773, 2.19862661617361079661334145404, 2.64139272167887492536232147129, 3.30078339073299021127094761027, 4.00517135852989304461611093037, 4.57730489606276588333642305309, 5.13364610600543206575722297600, 5.50325664724285495618660757850, 6.02136980541090391017209854900, 6.66802986371476306529165742036, 7.16821278301953870536747588848, 7.55756739866551106279076578094, 7.933617314069736134368829467402, 8.613939219671200765460315094925, 9.026129019908772187071485135707
