| L(s) = 1 | + 2·2-s − 7.52·3-s + 4·4-s − 15.0·6-s − 14·7-s + 8·8-s + 29.5·9-s + 27.1·11-s − 30.0·12-s + 7.52·13-s − 28·14-s + 16·16-s − 17·17-s + 59.1·18-s − 43.7·19-s + 105.·21-s + 54.2·22-s + 88.4·23-s − 60.1·24-s + 15.0·26-s − 19.2·27-s − 56·28-s − 27.5·29-s + 197.·31-s + 32·32-s − 204·33-s − 34·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.44·3-s + 0.5·4-s − 1.02·6-s − 0.755·7-s + 0.353·8-s + 1.09·9-s + 0.743·11-s − 0.723·12-s + 0.160·13-s − 0.534·14-s + 0.250·16-s − 0.242·17-s + 0.774·18-s − 0.528·19-s + 1.09·21-s + 0.525·22-s + 0.801·23-s − 0.511·24-s + 0.113·26-s − 0.137·27-s − 0.377·28-s − 0.176·29-s + 1.14·31-s + 0.176·32-s − 1.07·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 7.52T + 27T^{2} \) |
| 7 | \( 1 + 14T + 343T^{2} \) |
| 11 | \( 1 - 27.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.52T + 2.19e3T^{2} \) |
| 19 | \( 1 + 43.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 88.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 435.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 76.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 654.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 179.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 473.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 298.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 993.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 44.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 446.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3T + 9.12e5T^{2} \) |
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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
−9.626359701815715973885565235834, −8.552356006664275302987793859610, −7.12532360846178535310766040772, −6.50501406180447932305749053920, −5.93760950708865348050808081353, −4.97077689885692375074416503145, −4.14251964157091445753837664890, −2.97453971693808804441772439120, −1.34242855480123506717498122659, 0,
1.34242855480123506717498122659, 2.97453971693808804441772439120, 4.14251964157091445753837664890, 4.97077689885692375074416503145, 5.93760950708865348050808081353, 6.50501406180447932305749053920, 7.12532360846178535310766040772, 8.552356006664275302987793859610, 9.626359701815715973885565235834
