| L(s) = 1 | + 47·4-s − 1.45e3·9-s + 3.92e3·11-s − 1.88e3·16-s + 4.24e4·29-s − 6.85e4·36-s + 1.84e5·44-s − 1.17e5·49-s − 2.81e5·64-s − 4.84e5·71-s − 1.85e6·79-s + 1.59e6·81-s − 5.72e6·99-s + 5.17e6·109-s + 1.99e6·116-s + 8.00e6·121-s + 127-s + 131-s + 137-s + 139-s + 2.75e6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.65e6·169-s + ⋯ |
| L(s) = 1 | + 0.734·4-s − 2·9-s + 2.94·11-s − 0.460·16-s + 1.74·29-s − 1.46·36-s + 2.16·44-s − 49-s − 1.07·64-s − 1.35·71-s − 3.76·79-s + 3·81-s − 5.89·99-s + 3.99·109-s + 1.27·116-s + 4.51·121-s + 0.921·144-s − 2·169-s − 1.35·176-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.098103943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.098103943\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 47 T^{2} + p^{12} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 1962 T + p^{6} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 220762978 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 21222 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 5108772818 T^{2} + p^{12} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3388378898 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 41794002542 T^{2} + p^{12} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 178008750862 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 242478 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 929378 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
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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
−11.74036935979655328959274647014, −11.36537578708700889400367122824, −11.29065284569425288043641070846, −10.41521622959393079911369002663, −9.834652668477942549405129683794, −9.168540547654333646543529353491, −8.715537448204691557805647060576, −8.621904808408269876392576066345, −7.76162746669072953731550823769, −6.80964819058357011010041364174, −6.76740520847528491400210075471, −5.99105910457661947598869054627, −5.84243057149328965670756079339, −4.67759109887024183184429657838, −4.22428792046001908466741510012, −3.25997374580471508682336098393, −2.95931332052088730767286080079, −2.00696305529256835684234165900, −1.32862181308718705808746466602, −0.51223629433448037460782274566,
0.51223629433448037460782274566, 1.32862181308718705808746466602, 2.00696305529256835684234165900, 2.95931332052088730767286080079, 3.25997374580471508682336098393, 4.22428792046001908466741510012, 4.67759109887024183184429657838, 5.84243057149328965670756079339, 5.99105910457661947598869054627, 6.76740520847528491400210075471, 6.80964819058357011010041364174, 7.76162746669072953731550823769, 8.621904808408269876392576066345, 8.715537448204691557805647060576, 9.168540547654333646543529353491, 9.834652668477942549405129683794, 10.41521622959393079911369002663, 11.29065284569425288043641070846, 11.36537578708700889400367122824, 11.74036935979655328959274647014
