| L(s) = 1 | + 968·7-s + 6.73e3·13-s − 1.14e4·19-s + 992·25-s + 7.95e4·31-s + 1.05e5·37-s − 7.60e3·43-s + 4.67e5·49-s + 2.65e4·61-s − 3.37e5·67-s + 4.72e5·73-s + 7.02e4·79-s + 6.52e6·91-s − 6.42e5·97-s − 3.98e6·103-s + 3.88e5·109-s + 1.74e6·121-s + 127-s + 131-s − 1.11e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2.82·7-s + 3.06·13-s − 1.67·19-s + 0.0634·25-s + 2.67·31-s + 2.07·37-s − 0.0955·43-s + 3.97·49-s + 0.116·61-s − 1.12·67-s + 1.21·73-s + 0.142·79-s + 8.65·91-s − 0.704·97-s − 3.64·103-s + 0.300·109-s + 0.985·121-s − 4.72·133-s + 5.05·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(6.302585493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.302585493\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 992 T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 484 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1745714 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3368 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 48274976 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5744 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 284666690 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 327940544 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 39796 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 52526 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8128061984 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3800 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 15661450658 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12666935680 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 21940729490 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13250 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 168968 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 26256724990 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 236144 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 35116 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 653760187346 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 977236742720 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 321424 T + p^{6} T^{2} )^{2} \) |
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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
−11.83797341295481120918366888217, −11.75658301839717524534462347246, −10.96299692797315305569050367123, −10.94325142230083535338597538769, −10.58971707827712798579318381850, −9.621784142988911707236896353435, −8.772853546570716716963485991316, −8.443101918666047310890345475419, −8.015378540875718728436061159478, −7.982442961510968947010264665721, −6.70100945721013363536436576813, −6.24352548407250608505165201742, −5.70754577173580074259172128053, −4.87842778730165009840672362270, −4.25270911390618045706173370341, −4.07536074456574609125939238883, −2.80971070822736746315287972799, −1.92145876805885855154170684455, −1.27129565460516531850455623138, −0.916004814440346501894964494509,
0.916004814440346501894964494509, 1.27129565460516531850455623138, 1.92145876805885855154170684455, 2.80971070822736746315287972799, 4.07536074456574609125939238883, 4.25270911390618045706173370341, 4.87842778730165009840672362270, 5.70754577173580074259172128053, 6.24352548407250608505165201742, 6.70100945721013363536436576813, 7.982442961510968947010264665721, 8.015378540875718728436061159478, 8.443101918666047310890345475419, 8.772853546570716716963485991316, 9.621784142988911707236896353435, 10.58971707827712798579318381850, 10.94325142230083535338597538769, 10.96299692797315305569050367123, 11.75658301839717524534462347246, 11.83797341295481120918366888217
