| L(s) = 1 | − 2-s − 3-s + 4-s − 1.23·5-s + 6-s + 2.50·7-s − 8-s + 9-s + 1.23·10-s + 2.68·11-s − 12-s − 6.45·13-s − 2.50·14-s + 1.23·15-s + 16-s + 7.36·17-s − 18-s + 19-s − 1.23·20-s − 2.50·21-s − 2.68·22-s − 1.51·23-s + 24-s − 3.47·25-s + 6.45·26-s − 27-s + 2.50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.553·5-s + 0.408·6-s + 0.945·7-s − 0.353·8-s + 0.333·9-s + 0.391·10-s + 0.809·11-s − 0.288·12-s − 1.79·13-s − 0.668·14-s + 0.319·15-s + 0.250·16-s + 1.78·17-s − 0.235·18-s + 0.229·19-s − 0.276·20-s − 0.545·21-s − 0.572·22-s − 0.316·23-s + 0.204·24-s − 0.694·25-s + 1.26·26-s − 0.192·27-s + 0.472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 + 6.45T + 13T^{2} \) |
| 17 | \( 1 - 7.36T + 17T^{2} \) |
| 23 | \( 1 + 1.51T + 23T^{2} \) |
| 29 | \( 1 + 8.34T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 - 4.05T + 37T^{2} \) |
| 41 | \( 1 + 8.17T + 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 59 | \( 1 + 9.31T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 - 5.63T + 71T^{2} \) |
| 73 | \( 1 + 4.46T + 73T^{2} \) |
| 79 | \( 1 + 2.74T + 79T^{2} \) |
| 83 | \( 1 - 9.55T + 83T^{2} \) |
| 89 | \( 1 + 0.527T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.86786635375892340288548812323, −7.23453632475523146046034168211, −6.44668548201832411395914965603, −5.50964064648166911658276364348, −4.96671892615677997004554447513, −4.09957015358873959929026609190, −3.19300440810838284823967113174, −2.01687412561450706298057131132, −1.18990905784933293822388014541, 0,
1.18990905784933293822388014541, 2.01687412561450706298057131132, 3.19300440810838284823967113174, 4.09957015358873959929026609190, 4.96671892615677997004554447513, 5.50964064648166911658276364348, 6.44668548201832411395914965603, 7.23453632475523146046034168211, 7.86786635375892340288548812323
