L(s) = 1 | − 2.46·2-s + 1.74·3-s + 4.08·4-s − 1.65·5-s − 4.29·6-s + 1.73·7-s − 5.13·8-s + 0.0297·9-s + 4.06·10-s − 0.125·11-s + 7.10·12-s − 4.53·13-s − 4.28·14-s − 2.87·15-s + 4.50·16-s − 4.82·17-s − 0.0734·18-s − 2.48·19-s − 6.73·20-s + 3.02·21-s + 0.309·22-s + 7.67·23-s − 8.94·24-s − 2.27·25-s + 11.1·26-s − 5.17·27-s + 7.09·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s + 1.00·3-s + 2.04·4-s − 0.737·5-s − 1.75·6-s + 0.656·7-s − 1.81·8-s + 0.00992·9-s + 1.28·10-s − 0.0378·11-s + 2.05·12-s − 1.25·13-s − 1.14·14-s − 0.741·15-s + 1.12·16-s − 1.17·17-s − 0.0173·18-s − 0.568·19-s − 1.50·20-s + 0.660·21-s + 0.0659·22-s + 1.60·23-s − 1.82·24-s − 0.455·25-s + 2.19·26-s − 0.994·27-s + 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7332485620\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7332485620\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 6037 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 1.74T + 3T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 0.125T + 11T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 23 | \( 1 - 7.67T + 23T^{2} \) |
| 29 | \( 1 - 8.80T + 29T^{2} \) |
| 31 | \( 1 - 3.57T + 31T^{2} \) |
| 37 | \( 1 - 0.599T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 + 2.63T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 9.52T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 0.474T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 9.87T + 79T^{2} \) |
| 83 | \( 1 + 1.54T + 83T^{2} \) |
| 89 | \( 1 - 0.501T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.225747872766836192299148975793, −7.78336802952673740857381752624, −6.95368908255518425795474229771, −6.57625809183711315629924700227, −5.08017510163692907574004576648, −4.41114393178753591210353490414, −3.19076932095083713355620175026, −2.49380595771328739646291823577, −1.82939197604960077933128778118, −0.53596872232720582644108570546,
0.53596872232720582644108570546, 1.82939197604960077933128778118, 2.49380595771328739646291823577, 3.19076932095083713355620175026, 4.41114393178753591210353490414, 5.08017510163692907574004576648, 6.57625809183711315629924700227, 6.95368908255518425795474229771, 7.78336802952673740857381752624, 8.225747872766836192299148975793