| L(s) = 1 | + 1.87·2-s − 3-s + 2.53·4-s − 1.87·6-s − 1.53·7-s + 2.87·8-s + 9-s + 0.347·11-s − 2.53·12-s + 1.87·13-s − 2.87·14-s + 2.87·16-s − 0.347·17-s + 1.87·18-s + 1.53·21-s + 0.652·22-s − 2.87·24-s + 3.53·26-s − 27-s − 3.87·28-s + 29-s + 2.53·32-s − 0.347·33-s − 0.652·34-s + 2.53·36-s − 1.87·39-s − 41-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 3-s + 2.53·4-s − 1.87·6-s − 1.53·7-s + 2.87·8-s + 9-s + 0.347·11-s − 2.53·12-s + 1.87·13-s − 2.87·14-s + 2.87·16-s − 0.347·17-s + 1.87·18-s + 1.53·21-s + 0.652·22-s − 2.87·24-s + 3.53·26-s − 27-s − 3.87·28-s + 29-s + 2.53·32-s − 0.347·33-s − 0.652·34-s + 2.53·36-s − 1.87·39-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.606967116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.606967116\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.87T + T^{2} \) |
| 7 | \( 1 + 1.53T + T^{2} \) |
| 11 | \( 1 - 0.347T + T^{2} \) |
| 13 | \( 1 - 1.87T + T^{2} \) |
| 17 | \( 1 + 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.53T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.87T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.526165874879018027358411812179, −8.292539683389288985112163405555, −6.91906838436844395268577660582, −6.57361497476501462714389800745, −6.07897706859304921688609895872, −5.37791348032863760307696527283, −4.35136885190179736969885903481, −3.69950209953707938992029285562, −2.98179984216700250557221153793, −1.47579939597712814270817544533,
1.47579939597712814270817544533, 2.98179984216700250557221153793, 3.69950209953707938992029285562, 4.35136885190179736969885903481, 5.37791348032863760307696527283, 6.07897706859304921688609895872, 6.57361497476501462714389800745, 6.91906838436844395268577660582, 8.292539683389288985112163405555, 9.526165874879018027358411812179
