| L(s) = 1 | − 2.64·2-s + 3-s + 4.97·4-s − 3.51·5-s − 2.64·6-s + 7-s − 7.87·8-s + 9-s + 9.28·10-s + 1.28·11-s + 4.97·12-s + 5.89·13-s − 2.64·14-s − 3.51·15-s + 10.8·16-s − 2.10·17-s − 2.64·18-s + 1.10·19-s − 17.5·20-s + 21-s − 3.40·22-s − 4.43·23-s − 7.87·24-s + 7.35·25-s − 15.5·26-s + 27-s + 4.97·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.577·3-s + 2.48·4-s − 1.57·5-s − 1.07·6-s + 0.377·7-s − 2.78·8-s + 0.333·9-s + 2.93·10-s + 0.388·11-s + 1.43·12-s + 1.63·13-s − 0.706·14-s − 0.907·15-s + 2.70·16-s − 0.510·17-s − 0.622·18-s + 0.254·19-s − 3.91·20-s + 0.218·21-s − 0.726·22-s − 0.924·23-s − 1.60·24-s + 1.47·25-s − 3.05·26-s + 0.192·27-s + 0.941·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 5 | \( 1 + 3.51T + 5T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 - 1.10T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 + 1.70T + 29T^{2} \) |
| 31 | \( 1 + 3.10T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 + 5.44T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 + 7.02T + 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 + 7.22T + 79T^{2} \) |
| 83 | \( 1 - 4.97T + 83T^{2} \) |
| 89 | \( 1 + 4.39T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−7.81778580367705528092614621056, −7.15551838032193586437955660182, −6.58087244397774077422016484167, −5.72317735336537028882894495321, −4.29781829430294420838241216343, −3.72285521535718411524781354174, −2.95592276168747681634985182301, −1.84221073879008699482390875085, −1.09510455473148841395314278659, 0,
1.09510455473148841395314278659, 1.84221073879008699482390875085, 2.95592276168747681634985182301, 3.72285521535718411524781354174, 4.29781829430294420838241216343, 5.72317735336537028882894495321, 6.58087244397774077422016484167, 7.15551838032193586437955660182, 7.81778580367705528092614621056
