| L(s) = 1 | − 3.16·2-s + 7.08·3-s + 1.99·4-s − 13.6·5-s − 22.4·6-s + 14.3·7-s + 18.9·8-s + 23.2·9-s + 43.0·10-s − 67.7·11-s + 14.1·12-s − 45.3·14-s − 96.4·15-s − 75.9·16-s + 0.337·17-s − 73.5·18-s + 40.5·19-s − 27.1·20-s + 101.·21-s + 214.·22-s − 155.·23-s + 134.·24-s + 60.0·25-s − 26.5·27-s + 28.5·28-s − 33.7·29-s + 304.·30-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 1.36·3-s + 0.249·4-s − 1.21·5-s − 1.52·6-s + 0.773·7-s + 0.839·8-s + 0.861·9-s + 1.35·10-s − 1.85·11-s + 0.340·12-s − 0.864·14-s − 1.65·15-s − 1.18·16-s + 0.00481·17-s − 0.962·18-s + 0.489·19-s − 0.303·20-s + 1.05·21-s + 2.07·22-s − 1.41·23-s + 1.14·24-s + 0.480·25-s − 0.189·27-s + 0.192·28-s − 0.216·29-s + 1.85·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 + 3.16T + 8T^{2} \) |
| 3 | \( 1 - 7.08T + 27T^{2} \) |
| 5 | \( 1 + 13.6T + 125T^{2} \) |
| 7 | \( 1 - 14.3T + 343T^{2} \) |
| 11 | \( 1 + 67.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 0.337T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 33.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 58.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 59.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 560.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 269.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 60.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 282.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 984.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 539.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.58e3T + 9.12e5T^{2} \) |
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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
−11.54401437994295894113742799497, −10.57100798982136116907291389734, −9.579507243656445362847698219557, −8.400869729525969421643765889863, −7.943332756351356512369300997241, −7.52144673820047665986124527512, −4.95965132993804535854587985402, −3.62027052683656365834395177214, −2.08378530066237003773905053901, 0,
2.08378530066237003773905053901, 3.62027052683656365834395177214, 4.95965132993804535854587985402, 7.52144673820047665986124527512, 7.943332756351356512369300997241, 8.400869729525969421643765889863, 9.579507243656445362847698219557, 10.57100798982136116907291389734, 11.54401437994295894113742799497
