| L(s) = 1 | − 5.17·2-s + 3·3-s + 18.8·4-s + 17.1·5-s − 15.5·6-s − 19.8·7-s − 56.0·8-s + 9·9-s − 88.5·10-s − 47.5·11-s + 56.4·12-s − 63.9·13-s + 103.·14-s + 51.3·15-s + 139.·16-s + 10.7·17-s − 46.6·18-s − 133.·19-s + 321.·20-s − 59.6·21-s + 246.·22-s + 122.·23-s − 168.·24-s + 167.·25-s + 331.·26-s + 27·27-s − 374.·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.577·3-s + 2.35·4-s + 1.52·5-s − 1.05·6-s − 1.07·7-s − 2.47·8-s + 0.333·9-s − 2.80·10-s − 1.30·11-s + 1.35·12-s − 1.36·13-s + 1.96·14-s + 0.883·15-s + 2.18·16-s + 0.153·17-s − 0.610·18-s − 1.61·19-s + 3.59·20-s − 0.620·21-s + 2.38·22-s + 1.11·23-s − 1.42·24-s + 1.33·25-s + 2.49·26-s + 0.192·27-s − 2.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 + 67T \) |
| good | 2 | \( 1 + 5.17T + 8T^{2} \) |
| 5 | \( 1 - 17.1T + 125T^{2} \) |
| 7 | \( 1 + 19.8T + 343T^{2} \) |
| 11 | \( 1 + 47.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 63.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 161.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 228.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 273.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 414.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 215.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 674.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 577.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 245.T + 2.26e5T^{2} \) |
| 71 | \( 1 + 325.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 285.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 298.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 92.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 871.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−10.73241792533453951022473987100, −10.23657302592555172563896128793, −9.426631815200028978984769700977, −8.917762391333367813245020228361, −7.56886969589852243784907360169, −6.75281102455023465571879884762, −5.54996425670707457018266274619, −2.74196089265084317987924167886, −2.02918787600409761608063260615, 0,
2.02918787600409761608063260615, 2.74196089265084317987924167886, 5.54996425670707457018266274619, 6.75281102455023465571879884762, 7.56886969589852243784907360169, 8.917762391333367813245020228361, 9.426631815200028978984769700977, 10.23657302592555172563896128793, 10.73241792533453951022473987100
