| L(s) = 1 | − 2.18e3·9-s − 1.56e5·25-s + 1.23e6·37-s + 1.25e6·43-s − 8.11e6·67-s − 8.49e6·79-s + 4.78e6·81-s − 5.34e7·109-s + 3.89e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.33e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 9-s − 2·25-s + 3.99·37-s + 2.40·43-s − 3.29·67-s − 1.93·79-s + 81-s − 3.95·109-s + 2·121-s + 0.532·169-s + 2·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8965658320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8965658320\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 12605 T + p^{7} T^{2} )( 1 + 12605 T + p^{7} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 43091 T + p^{7} T^{2} )( 1 + 43091 T + p^{7} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 152471 T + p^{7} T^{2} )( 1 + 152471 T + p^{7} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 615373 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 625729 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3535546 T + p^{7} T^{2} )( 1 + 3535546 T + p^{7} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4058455 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5038001 T + p^{7} T^{2} )( 1 + 5038001 T + p^{7} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4245427 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12245198 T + p^{7} T^{2} )( 1 + 12245198 T + p^{7} T^{2} ) \) |
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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
−10.03422812120967476129010004060, −9.366597620313767245299712982764, −8.919257746833733149560681232022, −8.550744120521715859356832288872, −7.85414816834067340010692403051, −7.59948805691415774659612022471, −7.45860667648896383567246172595, −6.35867661056366441446976478321, −6.23101977248343311941517683903, −5.64124564427981927054078240803, −5.56996622187949316599266306439, −4.43741672635380867220354686603, −4.37698722018241803269034327801, −3.81581323155376297421657177850, −2.99283343644449877537428263672, −2.64140732442430584944111883095, −2.24819636837214966581380818194, −1.40320438141071576707233008059, −0.912020959248493948185389828580, −0.20108071525594844312480879685,
0.20108071525594844312480879685, 0.912020959248493948185389828580, 1.40320438141071576707233008059, 2.24819636837214966581380818194, 2.64140732442430584944111883095, 2.99283343644449877537428263672, 3.81581323155376297421657177850, 4.37698722018241803269034327801, 4.43741672635380867220354686603, 5.56996622187949316599266306439, 5.64124564427981927054078240803, 6.23101977248343311941517683903, 6.35867661056366441446976478321, 7.45860667648896383567246172595, 7.59948805691415774659612022471, 7.85414816834067340010692403051, 8.550744120521715859356832288872, 8.919257746833733149560681232022, 9.366597620313767245299712982764, 10.03422812120967476129010004060
