| L(s) = 1 | + 0.430·2-s − 1.25·3-s − 1.81·4-s + 5-s − 0.542·6-s + 4.34·7-s − 1.64·8-s − 1.41·9-s + 0.430·10-s + 11-s + 2.28·12-s + 2.04·13-s + 1.87·14-s − 1.25·15-s + 2.92·16-s − 4.46·17-s − 0.609·18-s − 0.805·19-s − 1.81·20-s − 5.47·21-s + 0.430·22-s − 4.40·23-s + 2.06·24-s + 25-s + 0.882·26-s + 5.55·27-s − 7.88·28-s + ⋯ |
| L(s) = 1 | + 0.304·2-s − 0.727·3-s − 0.907·4-s + 0.447·5-s − 0.221·6-s + 1.64·7-s − 0.581·8-s − 0.471·9-s + 0.136·10-s + 0.301·11-s + 0.659·12-s + 0.567·13-s + 0.500·14-s − 0.325·15-s + 0.730·16-s − 1.08·17-s − 0.143·18-s − 0.184·19-s − 0.405·20-s − 1.19·21-s + 0.0918·22-s − 0.917·23-s + 0.422·24-s + 0.200·25-s + 0.172·26-s + 1.06·27-s − 1.48·28-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 | \( 1 - 0.430T + 2T^{2} \) |
| 3 | \( 1 + 1.25T + 3T^{2} \) |
| 7 | \( 1 - 4.34T + 7T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 + 0.805T + 19T^{2} \) |
| 23 | \( 1 + 4.40T + 23T^{2} \) |
| 29 | \( 1 + 8.69T + 29T^{2} \) |
| 31 | \( 1 + 8.06T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 - 3.55T + 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 0.726T + 67T^{2} \) |
| 71 | \( 1 + 1.00T + 71T^{2} \) |
| 79 | \( 1 - 3.36T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 0.909T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−8.202299094646925982481832600148, −7.39791928644836844832196052691, −6.30647180327293575716134019140, −5.58798569073376960138355902707, −5.25415584464259385823454136819, −4.34531199958828622926991285438, −3.79698455898855723789704027415, −2.31706420325243333155503028364, −1.37418569740322698034632566844, 0,
1.37418569740322698034632566844, 2.31706420325243333155503028364, 3.79698455898855723789704027415, 4.34531199958828622926991285438, 5.25415584464259385823454136819, 5.58798569073376960138355902707, 6.30647180327293575716134019140, 7.39791928644836844832196052691, 8.202299094646925982481832600148
