| L(s) = 1 | + 508·7-s + 1.46e4·13-s − 1.14e5·19-s + 7.81e4·25-s − 1.78e5·31-s + 5.59e5·37-s − 1.03e6·43-s + 8.23e5·49-s + 3.53e6·61-s + 3.85e5·67-s − 1.25e7·73-s − 8.76e6·79-s + 7.42e6·91-s − 1.22e7·97-s − 8.02e6·103-s − 3.36e7·109-s + 1.94e7·121-s + 127-s + 131-s − 5.83e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.559·7-s + 1.84·13-s − 3.84·19-s + 25-s − 1.07·31-s + 1.81·37-s − 1.98·43-s + 49-s + 1.99·61-s + 0.156·67-s − 3.77·73-s − 1.99·79-s + 1.03·91-s − 1.36·97-s − 0.723·103-s − 2.48·109-s + 121-s − 2.15·133-s + 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5980364912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5980364912\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 1763 T + p^{7} T^{2} )( 1 + 1255 T + p^{7} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 12605 T + p^{7} T^{2} )( 1 - 2009 T + p^{7} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 57448 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 152471 T + p^{7} T^{2} )( 1 + 331387 T + p^{7} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 279710 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 409495 T + p^{7} T^{2} )( 1 + 625729 T + p^{7} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1998347 T + p^{7} T^{2} )( 1 - 1537199 T + p^{7} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4443527 T + p^{7} T^{2} )( 1 + 4058455 T + p^{7} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6274810 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4245427 T + p^{7} T^{2} )( 1 + 4517617 T + p^{7} T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5276357 T + p^{7} T^{2} )( 1 + 17521555 T + p^{7} T^{2} ) \) |
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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
−10.95119745631876226167027558609, −10.29940363016224894912420686312, −9.834554280364640273152137670199, −8.826140585061420586479154876158, −8.796756336273071924717728165691, −8.408070549404647864915456561955, −8.058120016828688944357972899758, −7.17782778053711466020887180993, −6.77035846040192467002643539159, −6.15505695424616754291157724793, −6.01744297277269847860418110028, −5.22272070923423882114175293939, −4.54766563850157560271388918364, −3.97071074778870702029558842940, −3.90101906820503823643514218606, −2.77367255910472664265726543727, −2.34901576041936390968817487368, −1.50310821646457663252130751021, −1.27426181892304569304764656119, −0.16543174898229926918656731752,
0.16543174898229926918656731752, 1.27426181892304569304764656119, 1.50310821646457663252130751021, 2.34901576041936390968817487368, 2.77367255910472664265726543727, 3.90101906820503823643514218606, 3.97071074778870702029558842940, 4.54766563850157560271388918364, 5.22272070923423882114175293939, 6.01744297277269847860418110028, 6.15505695424616754291157724793, 6.77035846040192467002643539159, 7.17782778053711466020887180993, 8.058120016828688944357972899758, 8.408070549404647864915456561955, 8.796756336273071924717728165691, 8.826140585061420586479154876158, 9.834554280364640273152137670199, 10.29940363016224894912420686312, 10.95119745631876226167027558609
