| L(s) = 1 | + 4.12·3-s − 10.8·5-s − 29.9·7-s − 9.95·9-s + 4.75·11-s − 75.3·13-s − 44.9·15-s + 111.·17-s + 90.8·19-s − 123.·21-s − 42.3·23-s − 6.36·25-s − 152.·27-s − 202.·29-s − 31·31-s + 19.6·33-s + 326.·35-s + 334.·37-s − 311.·39-s − 367.·41-s + 82.6·43-s + 108.·45-s − 276.·47-s + 556.·49-s + 461.·51-s + 581.·53-s − 51.7·55-s + ⋯ |
| L(s) = 1 | + 0.794·3-s − 0.974·5-s − 1.61·7-s − 0.368·9-s + 0.130·11-s − 1.60·13-s − 0.774·15-s + 1.59·17-s + 1.09·19-s − 1.28·21-s − 0.383·23-s − 0.0509·25-s − 1.08·27-s − 1.29·29-s − 0.179·31-s + 0.103·33-s + 1.57·35-s + 1.48·37-s − 1.27·39-s − 1.39·41-s + 0.293·43-s + 0.359·45-s − 0.858·47-s + 1.62·49-s + 1.26·51-s + 1.50·53-s − 0.126·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + 31T \) |
| good | 3 | \( 1 - 4.12T + 27T^{2} \) |
| 5 | \( 1 + 10.8T + 125T^{2} \) |
| 7 | \( 1 + 29.9T + 343T^{2} \) |
| 11 | \( 1 - 4.75T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 42.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 202.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 334.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 367.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 82.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 581.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 630.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 527.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 684.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 230.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 802.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 455.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 214.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 340.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 995.T + 9.12e5T^{2} \) |
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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
−12.35896192425059518299990958034, −11.68914747985842929624541501773, −9.908331355948405768547495236472, −9.425497500696559711852702988877, −7.935216206166005237899942995115, −7.22827018453745253974585434252, −5.61105462460997553249297717092, −3.71187779542170186382938953812, −2.88383176420668570512195588458, 0,
2.88383176420668570512195588458, 3.71187779542170186382938953812, 5.61105462460997553249297717092, 7.22827018453745253974585434252, 7.935216206166005237899942995115, 9.425497500696559711852702988877, 9.908331355948405768547495236472, 11.68914747985842929624541501773, 12.35896192425059518299990958034
