| L(s) = 1 | + 9i·2-s − 17·4-s − 343i·7-s + 423i·8-s − 729·9-s + 1.96e3·11-s + 3.08e3·14-s − 4.89e3·16-s − 6.56e3i·18-s + 1.76e4i·22-s + 2.27e4i·23-s + 5.83e3i·28-s + 2.12e4·29-s − 1.69e4i·32-s + 1.23e4·36-s + 1.01e5i·37-s + ⋯ |
| L(s) = 1 | + 1.12i·2-s − 0.265·4-s − i·7-s + 0.826i·8-s − 0.999·9-s + 1.47·11-s + 1.12·14-s − 1.19·16-s − 1.12i·18-s + 1.65i·22-s + 1.86i·23-s + 0.265i·28-s + 0.870·29-s − 0.518i·32-s + 0.265·36-s + 1.99i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.760143159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.760143159\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 343iT \) |
| good | 2 | \( 1 - 9iT - 64T^{2} \) |
| 3 | \( 1 + 729T^{2} \) |
| 11 | \( 1 - 1.96e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 4.82e6T^{2} \) |
| 17 | \( 1 + 2.41e7T^{2} \) |
| 19 | \( 1 - 4.70e7T^{2} \) |
| 23 | \( 1 - 2.27e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.12e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.01e5iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.26e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.07e10T^{2} \) |
| 53 | \( 1 + 5.03e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 - 5.15e10T^{2} \) |
| 67 | \( 1 + 5.39e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.42e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 1.51e11T^{2} \) |
| 79 | \( 1 + 9.29e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.26e11T^{2} \) |
| 89 | \( 1 - 4.96e11T^{2} \) |
| 97 | \( 1 + 8.32e11T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.74036935979655328959274647014, −11.29065284569425288043641070846, −9.834652668477942549405129683794, −8.715537448204691557805647060576, −7.76162746669072953731550823769, −6.76740520847528491400210075471, −5.99105910457661947598869054627, −4.67759109887024183184429657838, −3.25997374580471508682336098393, −1.32862181308718705808746466602,
0.51223629433448037460782274566, 2.00696305529256835684234165900, 2.95931332052088730767286080079, 4.22428792046001908466741510012, 5.84243057149328965670756079339, 6.80964819058357011010041364174, 8.621904808408269876392576066345, 9.168540547654333646543529353491, 10.41521622959393079911369002663, 11.36537578708700889400367122824
