| L(s) = 1 | − 4.10·2-s − 1.59·3-s + 8.81·4-s − 13.7·5-s + 6.52·6-s − 19.9·7-s − 3.35·8-s − 24.4·9-s + 56.3·10-s − 11·11-s − 14.0·12-s − 28.3·13-s + 81.8·14-s + 21.8·15-s − 56.7·16-s − 17·17-s + 100.·18-s − 103.·19-s − 121.·20-s + 31.7·21-s + 45.1·22-s − 49.0·23-s + 5.33·24-s + 63.5·25-s + 116.·26-s + 81.9·27-s − 175.·28-s + ⋯ |
| L(s) = 1 | − 1.44·2-s − 0.306·3-s + 1.10·4-s − 1.22·5-s + 0.444·6-s − 1.07·7-s − 0.148·8-s − 0.906·9-s + 1.78·10-s − 0.301·11-s − 0.337·12-s − 0.604·13-s + 1.56·14-s + 0.376·15-s − 0.887·16-s − 0.242·17-s + 1.31·18-s − 1.24·19-s − 1.35·20-s + 0.330·21-s + 0.437·22-s − 0.444·23-s + 0.0453·24-s + 0.508·25-s + 0.875·26-s + 0.583·27-s − 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1368775724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1368775724\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 4.10T + 8T^{2} \) |
| 3 | \( 1 + 1.59T + 27T^{2} \) |
| 5 | \( 1 + 13.7T + 125T^{2} \) |
| 7 | \( 1 + 19.9T + 343T^{2} \) |
| 13 | \( 1 + 28.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 197.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 18.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 12.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 251.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 452.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 16.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 832.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 301.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 273.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 486.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 136.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−11.86403824926670091693908677817, −10.91610960366904134674934289956, −10.14101109853644760849638368562, −8.988124630327702236331172238366, −8.229864942105545091832950417844, −7.27458337997508901275554771829, −6.20378500639740975847840199485, −4.36476721258282919093992055251, −2.72148471516971976610988405475, −0.33627014160597585055485161484,
0.33627014160597585055485161484, 2.72148471516971976610988405475, 4.36476721258282919093992055251, 6.20378500639740975847840199485, 7.27458337997508901275554771829, 8.229864942105545091832950417844, 8.988124630327702236331172238366, 10.14101109853644760849638368562, 10.91610960366904134674934289956, 11.86403824926670091693908677817
