| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 12-s + 2·13-s − 14-s + 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 21-s + 24-s + 25-s + 2·26-s + 27-s − 28-s − 6·29-s + 30-s − 4·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.80266570790266, −13.52763387775028, −12.99867405100480, −12.48962787780858, −12.18275526946807, −11.49057647210868, −10.99662710459302, −10.53240039448314, −9.861330783895507, −9.445189203135269, −9.230872202870676, −8.222014281006107, −7.984218258875144, −7.365248428137021, −6.877199400910960, −6.220456904545868, −5.805144745809222, −5.210404925300091, −4.823954785448010, −3.867313789039899, −3.505062862245949, −3.092319934970367, −2.453551693600584, −1.500192814187600, −1.341146787755275, 0,
1.341146787755275, 1.500192814187600, 2.453551693600584, 3.092319934970367, 3.505062862245949, 3.867313789039899, 4.823954785448010, 5.210404925300091, 5.805144745809222, 6.220456904545868, 6.877199400910960, 7.365248428137021, 7.984218258875144, 8.222014281006107, 9.230872202870676, 9.445189203135269, 9.861330783895507, 10.53240039448314, 10.99662710459302, 11.49057647210868, 12.18275526946807, 12.48962787780858, 12.99867405100480, 13.52763387775028, 13.80266570790266
