L(s) = 1 | − 2-s − 2.52·3-s + 4-s − 1.30·5-s + 2.52·6-s + 0.799·7-s − 8-s + 3.35·9-s + 1.30·10-s − 2.77·11-s − 2.52·12-s + 3.89·13-s − 0.799·14-s + 3.29·15-s + 16-s + 7.65·17-s − 3.35·18-s − 19-s − 1.30·20-s − 2.01·21-s + 2.77·22-s − 0.364·23-s + 2.52·24-s − 3.29·25-s − 3.89·26-s − 0.902·27-s + 0.799·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.45·3-s + 0.5·4-s − 0.583·5-s + 1.02·6-s + 0.302·7-s − 0.353·8-s + 1.11·9-s + 0.412·10-s − 0.837·11-s − 0.727·12-s + 1.08·13-s − 0.213·14-s + 0.849·15-s + 0.250·16-s + 1.85·17-s − 0.791·18-s − 0.229·19-s − 0.291·20-s − 0.439·21-s + 0.592·22-s − 0.0759·23-s + 0.514·24-s − 0.659·25-s − 0.764·26-s − 0.173·27-s + 0.151·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 0.799T + 7T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 13 | \( 1 - 3.89T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 23 | \( 1 + 0.364T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 + 6.08T + 31T^{2} \) |
| 37 | \( 1 + 0.605T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 - 4.28T + 47T^{2} \) |
| 53 | \( 1 - 4.28T + 53T^{2} \) |
| 59 | \( 1 - 3.47T + 59T^{2} \) |
| 61 | \( 1 - 0.0120T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 5.61T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 5.00T + 97T^{2} \) |
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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
−7.46424620364377761494563504419, −6.87526033487713369256812118229, −6.01307778728579091842891273604, −5.52694296410180698559417127941, −4.96881023153971304838682681540, −3.86779073930678690252207964538, −3.20118471165529689520266650302, −1.82004218632450635531285194376, −0.943095196483826179719305998639, 0,
0.943095196483826179719305998639, 1.82004218632450635531285194376, 3.20118471165529689520266650302, 3.86779073930678690252207964538, 4.96881023153971304838682681540, 5.52694296410180698559417127941, 6.01307778728579091842891273604, 6.87526033487713369256812118229, 7.46424620364377761494563504419