| L(s) = 1 | − 27·3-s + 110·5-s − 504·7-s + 729·9-s − 3.81e3·11-s + 9.57e3·13-s − 2.97e3·15-s + 2.60e4·17-s + 3.83e4·19-s + 1.36e4·21-s + 7.11e4·23-s − 6.60e4·25-s − 1.96e4·27-s + 7.42e4·29-s + 2.75e5·31-s + 1.02e5·33-s − 5.54e4·35-s − 2.66e5·37-s − 2.58e5·39-s + 6.84e5·41-s − 2.45e5·43-s + 8.01e4·45-s − 4.78e5·47-s − 5.69e5·49-s − 7.04e5·51-s − 5.69e5·53-s − 4.19e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.393·5-s − 0.555·7-s + 1/3·9-s − 0.863·11-s + 1.20·13-s − 0.227·15-s + 1.28·17-s + 1.28·19-s + 0.320·21-s + 1.21·23-s − 0.845·25-s − 0.192·27-s + 0.565·29-s + 1.66·31-s + 0.498·33-s − 0.218·35-s − 0.865·37-s − 0.697·39-s + 1.55·41-s − 0.471·43-s + 0.131·45-s − 0.672·47-s − 0.691·49-s − 0.743·51-s − 0.525·53-s − 0.339·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.571113140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.571113140\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 - 22 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 72 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 3812 T + p^{7} T^{2} \) |
| 13 | \( 1 - 9574 T + p^{7} T^{2} \) |
| 17 | \( 1 - 26098 T + p^{7} T^{2} \) |
| 19 | \( 1 - 38308 T + p^{7} T^{2} \) |
| 23 | \( 1 - 71128 T + p^{7} T^{2} \) |
| 29 | \( 1 - 74262 T + p^{7} T^{2} \) |
| 31 | \( 1 - 275680 T + p^{7} T^{2} \) |
| 37 | \( 1 + 266610 T + p^{7} T^{2} \) |
| 41 | \( 1 - 684762 T + p^{7} T^{2} \) |
| 43 | \( 1 + 245956 T + p^{7} T^{2} \) |
| 47 | \( 1 + 478800 T + p^{7} T^{2} \) |
| 53 | \( 1 + 569410 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1525324 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2640458 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1416236 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3511304 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4738618 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4661488 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5729252 T + p^{7} T^{2} \) |
| 89 | \( 1 - 11993514 T + p^{7} T^{2} \) |
| 97 | \( 1 - 7150754 T + p^{7} T^{2} \) |
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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
−13.93912027478761531519888711761, −13.03938252752716325050772887124, −11.82402408992986872298267928579, −10.55633854013477313583727865023, −9.551456835265644911062996578868, −7.88232313545988823254620636967, −6.33582670566759877613820512821, −5.21450916606610879526421216009, −3.20451143351387760499013199530, −1.00113140934448355247805870095,
1.00113140934448355247805870095, 3.20451143351387760499013199530, 5.21450916606610879526421216009, 6.33582670566759877613820512821, 7.88232313545988823254620636967, 9.551456835265644911062996578868, 10.55633854013477313583727865023, 11.82402408992986872298267928579, 13.03938252752716325050772887124, 13.93912027478761531519888711761
