| L(s) = 1 | − 0.455·2-s + 3-s − 1.79·4-s − 0.749·5-s − 0.455·6-s − 7-s + 1.72·8-s + 9-s + 0.341·10-s − 1.95·11-s − 1.79·12-s + 2.51·13-s + 0.455·14-s − 0.749·15-s + 2.79·16-s − 1.75·17-s − 0.455·18-s − 3.76·19-s + 1.34·20-s − 21-s + 0.892·22-s + 7.74·23-s + 1.72·24-s − 4.43·25-s − 1.14·26-s + 27-s + 1.79·28-s + ⋯ |
| L(s) = 1 | − 0.322·2-s + 0.577·3-s − 0.896·4-s − 0.335·5-s − 0.185·6-s − 0.377·7-s + 0.610·8-s + 0.333·9-s + 0.107·10-s − 0.590·11-s − 0.517·12-s + 0.698·13-s + 0.121·14-s − 0.193·15-s + 0.699·16-s − 0.426·17-s − 0.107·18-s − 0.864·19-s + 0.300·20-s − 0.218·21-s + 0.190·22-s + 1.61·23-s + 0.352·24-s − 0.887·25-s − 0.224·26-s + 0.192·27-s + 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.455T + 2T^{2} \) |
| 5 | \( 1 + 0.749T + 5T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 + 1.75T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 29 | \( 1 - 1.80T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 + 8.09T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 3.25T + 47T^{2} \) |
| 53 | \( 1 - 0.297T + 53T^{2} \) |
| 59 | \( 1 - 0.440T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 1.44T + 71T^{2} \) |
| 73 | \( 1 + 2.90T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 9.80T + 97T^{2} \) |
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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
−8.442349031355814929832302466568, −8.078861864848315520165856273787, −7.06887112054013481860270902643, −6.29887149161716713554967289575, −5.14614516560716076386164665063, −4.45369367038046050639517480262, −3.61182968587681792949654632151, −2.77116511797731268744927019967, −1.38617598586779121683507745130, 0,
1.38617598586779121683507745130, 2.77116511797731268744927019967, 3.61182968587681792949654632151, 4.45369367038046050639517480262, 5.14614516560716076386164665063, 6.29887149161716713554967289575, 7.06887112054013481860270902643, 8.078861864848315520165856273787, 8.442349031355814929832302466568
