| L(s) = 1 | + 0.315·2-s + 2.38·3-s − 1.90·4-s − 5-s + 0.751·6-s + 0.00485·7-s − 1.23·8-s + 2.68·9-s − 0.315·10-s + 11-s − 4.53·12-s + 0.139·13-s + 0.00153·14-s − 2.38·15-s + 3.41·16-s − 1.09·17-s + 0.846·18-s − 4.53·19-s + 1.90·20-s + 0.0115·21-s + 0.315·22-s + 1.37·23-s − 2.93·24-s + 25-s + 0.0439·26-s − 0.755·27-s − 0.00922·28-s + ⋯ |
| L(s) = 1 | + 0.223·2-s + 1.37·3-s − 0.950·4-s − 0.447·5-s + 0.306·6-s + 0.00183·7-s − 0.434·8-s + 0.894·9-s − 0.0997·10-s + 0.301·11-s − 1.30·12-s + 0.0386·13-s + 0.000409·14-s − 0.615·15-s + 0.853·16-s − 0.264·17-s + 0.199·18-s − 1.04·19-s + 0.424·20-s + 0.00252·21-s + 0.0672·22-s + 0.287·23-s − 0.598·24-s + 0.200·25-s + 0.00862·26-s − 0.145·27-s − 0.00174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 0.315T + 2T^{2} \) |
| 3 | \( 1 - 2.38T + 3T^{2} \) |
| 7 | \( 1 - 0.00485T + 7T^{2} \) |
| 13 | \( 1 - 0.139T + 13T^{2} \) |
| 17 | \( 1 + 1.09T + 17T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 + 9.29T + 29T^{2} \) |
| 31 | \( 1 + 2.10T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 5.58T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 + 4.55T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 5.74T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 79 | \( 1 + 2.38T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 0.283T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.020023822383161619449087461714, −7.80563048294021688445465847031, −6.68343593754678752458194718112, −5.81275779152066254695096616437, −4.76711444541875005059609963027, −4.09427823542646371276644848729, −3.52084596129625850321363756415, −2.70194069479154922302279402532, −1.59889214887819480405469888824, 0,
1.59889214887819480405469888824, 2.70194069479154922302279402532, 3.52084596129625850321363756415, 4.09427823542646371276644848729, 4.76711444541875005059609963027, 5.81275779152066254695096616437, 6.68343593754678752458194718112, 7.80563048294021688445465847031, 8.020023822383161619449087461714
