| L(s) = 1 | + 4·3-s − 12·7-s + 7·9-s − 32·13-s + 4·19-s − 48·21-s − 5·25-s − 8·27-s − 36·31-s + 32·37-s − 128·39-s − 32·43-s + 10·49-s + 16·57-s − 164·61-s − 84·63-s − 48·67-s − 148·73-s − 20·75-s + 276·79-s − 95·81-s + 384·91-s − 144·93-s − 332·97-s + 52·103-s − 76·109-s + 128·111-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 1.71·7-s + 7/9·9-s − 2.46·13-s + 4/19·19-s − 2.28·21-s − 1/5·25-s − 0.296·27-s − 1.16·31-s + 0.864·37-s − 3.28·39-s − 0.744·43-s + 0.204·49-s + 0.280·57-s − 2.68·61-s − 4/3·63-s − 0.716·67-s − 2.02·73-s − 0.266·75-s + 3.49·79-s − 1.17·81-s + 4.21·91-s − 1.54·93-s − 3.42·97-s + 0.504·103-s − 0.697·109-s + 1.15·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1007212328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1007212328\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - 4 T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 222 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 558 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )( 1 + 44 T + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 702 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 558 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1998 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5598 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6942 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5598 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 138 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4958 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 166 T + p^{2} T^{2} )^{2} \) |
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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
−10.10072915033844897428547874623, −9.632789661362157763880760345370, −9.369317987145357652346132520001, −8.909903037297951816079553736473, −8.453390269387523133897739370828, −7.75737669104397396143380662810, −7.64047378899354147729507971155, −7.15067921766866296835217335077, −6.87044966700301108406721892460, −6.05577318849746578906627147384, −6.02148426302105952664043032474, −4.93971733998390532127082058106, −4.93326657896860365665155192059, −4.05494594856040619999608653792, −3.60653301888819599684046253340, −3.00132364844718114503731334732, −2.79605874642251400436888695018, −2.25058790495281515450366470130, −1.53133951491367683391982936359, −0.086027512788810667266821994792,
0.086027512788810667266821994792, 1.53133951491367683391982936359, 2.25058790495281515450366470130, 2.79605874642251400436888695018, 3.00132364844718114503731334732, 3.60653301888819599684046253340, 4.05494594856040619999608653792, 4.93326657896860365665155192059, 4.93971733998390532127082058106, 6.02148426302105952664043032474, 6.05577318849746578906627147384, 6.87044966700301108406721892460, 7.15067921766866296835217335077, 7.64047378899354147729507971155, 7.75737669104397396143380662810, 8.453390269387523133897739370828, 8.909903037297951816079553736473, 9.369317987145357652346132520001, 9.632789661362157763880760345370, 10.10072915033844897428547874623
