| L(s) = 1 | − 27·3-s + 240·5-s + 729·9-s + 702·11-s + 3.95e3·13-s − 6.48e3·15-s + 3.40e3·17-s + 4.90e4·19-s − 1.15e4·23-s − 2.05e4·25-s − 1.96e4·27-s + 4.96e4·29-s + 1.13e5·31-s − 1.89e4·33-s − 6.68e4·37-s − 1.06e5·39-s + 3.60e5·41-s − 7.65e5·43-s + 1.74e5·45-s + 1.34e6·47-s − 9.20e4·51-s + 3.58e5·53-s + 1.68e5·55-s − 1.32e6·57-s − 9.30e5·59-s + 1.31e6·61-s + 9.49e5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.858·5-s + 1/3·9-s + 0.159·11-s + 0.499·13-s − 0.495·15-s + 0.168·17-s + 1.64·19-s − 0.197·23-s − 0.262·25-s − 0.192·27-s + 0.378·29-s + 0.683·31-s − 0.0918·33-s − 0.217·37-s − 0.288·39-s + 0.817·41-s − 1.46·43-s + 0.286·45-s + 1.88·47-s − 0.0971·51-s + 0.331·53-s + 0.136·55-s − 0.946·57-s − 0.589·59-s + 0.743·61-s + 0.429·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.578013730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.578013730\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 48 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 702 T + p^{7} T^{2} \) |
| 13 | \( 1 - 3958 T + p^{7} T^{2} \) |
| 17 | \( 1 - 3408 T + p^{7} T^{2} \) |
| 19 | \( 1 - 49036 T + p^{7} T^{2} \) |
| 23 | \( 1 + 11514 T + p^{7} T^{2} \) |
| 29 | \( 1 - 49662 T + p^{7} T^{2} \) |
| 31 | \( 1 - 113320 T + p^{7} T^{2} \) |
| 37 | \( 1 + 66886 T + p^{7} T^{2} \) |
| 41 | \( 1 - 360900 T + p^{7} T^{2} \) |
| 43 | \( 1 + 765292 T + p^{7} T^{2} \) |
| 47 | \( 1 - 1344876 T + p^{7} T^{2} \) |
| 53 | \( 1 - 358962 T + p^{7} T^{2} \) |
| 59 | \( 1 + 930528 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1318834 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1893464 T + p^{7} T^{2} \) |
| 71 | \( 1 - 227994 T + p^{7} T^{2} \) |
| 73 | \( 1 + 784934 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2100892 T + p^{7} T^{2} \) |
| 83 | \( 1 + 8629308 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5903100 T + p^{7} T^{2} \) |
| 97 | \( 1 + 773846 T + p^{7} T^{2} \) |
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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
−9.756388597051748290697582849575, −8.827752786250166865787844611607, −7.71054123356580167003372963519, −6.74941074732568566764367685465, −5.85077608624838819198155766607, −5.25138465280012510175475516832, −4.04693593243787682732662094331, −2.85005310292873509105829673274, −1.60696949646246065145960734812, −0.74817975083930602558083296324,
0.74817975083930602558083296324, 1.60696949646246065145960734812, 2.85005310292873509105829673274, 4.04693593243787682732662094331, 5.25138465280012510175475516832, 5.85077608624838819198155766607, 6.74941074732568566764367685465, 7.71054123356580167003372963519, 8.827752786250166865787844611607, 9.756388597051748290697582849575
