| L(s) = 1 | + 1.58·2-s − 5.24·3-s − 5.48·4-s − 8.31·6-s − 21.3·8-s + 0.485·9-s + 28.1·11-s + 28.7·12-s − 3.85·13-s + 9.97·16-s + 38.3·17-s + 0.769·18-s − 116.·19-s + 44.6·22-s + 176.·23-s + 112.·24-s − 6.11·26-s + 139.·27-s − 209.·29-s + 207.·31-s + 186.·32-s − 147.·33-s + 60.8·34-s − 2.66·36-s + 15.6·37-s − 185.·38-s + 20.2·39-s + ⋯ |
| L(s) = 1 | + 0.560·2-s − 1.00·3-s − 0.685·4-s − 0.565·6-s − 0.945·8-s + 0.0179·9-s + 0.771·11-s + 0.691·12-s − 0.0823·13-s + 0.155·16-s + 0.547·17-s + 0.0100·18-s − 1.40·19-s + 0.432·22-s + 1.59·23-s + 0.953·24-s − 0.0461·26-s + 0.990·27-s − 1.34·29-s + 1.20·31-s + 1.03·32-s − 0.778·33-s + 0.306·34-s − 0.0123·36-s + 0.0696·37-s − 0.790·38-s + 0.0830·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.58T + 8T^{2} \) |
| 3 | \( 1 + 5.24T + 27T^{2} \) |
| 11 | \( 1 - 28.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.85T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 209.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 15.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 10.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 275.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 43.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 855.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 252.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 922.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 960.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 133.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−8.991883482780932734034687945729, −8.250920031237438189319274169809, −6.97091341450238588469731851970, −6.19000440966675608459251224426, −5.51977914047210016536911375934, −4.70945227456852018465084283379, −3.94481526772221748160289105920, −2.81208633668835677298735323334, −1.09456538326167978004730529567, 0,
1.09456538326167978004730529567, 2.81208633668835677298735323334, 3.94481526772221748160289105920, 4.70945227456852018465084283379, 5.51977914047210016536911375934, 6.19000440966675608459251224426, 6.97091341450238588469731851970, 8.250920031237438189319274169809, 8.991883482780932734034687945729
