| L(s) = 1 | + 2.62·2-s − 2.78·3-s + 4.87·4-s − 5-s − 7.29·6-s − 7-s + 7.53·8-s + 4.74·9-s − 2.62·10-s + 1.67·11-s − 13.5·12-s − 3.13·13-s − 2.62·14-s + 2.78·15-s + 10.0·16-s + 1.47·17-s + 12.4·18-s − 3.80·19-s − 4.87·20-s + 2.78·21-s + 4.37·22-s − 0.751·23-s − 20.9·24-s + 25-s − 8.21·26-s − 4.86·27-s − 4.87·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 1.60·3-s + 2.43·4-s − 0.447·5-s − 2.97·6-s − 0.377·7-s + 2.66·8-s + 1.58·9-s − 0.829·10-s + 0.503·11-s − 3.91·12-s − 0.869·13-s − 0.700·14-s + 0.718·15-s + 2.50·16-s + 0.358·17-s + 2.93·18-s − 0.872·19-s − 1.08·20-s + 0.607·21-s + 0.933·22-s − 0.156·23-s − 4.28·24-s + 0.200·25-s − 1.61·26-s − 0.937·27-s − 0.921·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 2.62T + 2T^{2} \) |
| 3 | \( 1 + 2.78T + 3T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 3.80T + 19T^{2} \) |
| 23 | \( 1 + 0.751T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 4.53T + 31T^{2} \) |
| 37 | \( 1 + 9.68T + 37T^{2} \) |
| 41 | \( 1 - 5.16T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.69T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 8.88T + 67T^{2} \) |
| 71 | \( 1 + 0.123T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 4.67T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.12590241915532987311224963918, −6.48009894063028604090247588733, −5.90200138624733794424773032535, −5.43884612324753441502059345887, −4.58698819090799228360140697923, −4.29479526888070971745231148640, −3.42592239670464563784901511859, −2.50422136631647884241725797725, −1.39760461031831221914104235684, 0,
1.39760461031831221914104235684, 2.50422136631647884241725797725, 3.42592239670464563784901511859, 4.29479526888070971745231148640, 4.58698819090799228360140697923, 5.43884612324753441502059345887, 5.90200138624733794424773032535, 6.48009894063028604090247588733, 7.12590241915532987311224963918
