| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 5·5-s − 6·6-s + 7-s − 8·8-s + 9·9-s + 10·10-s − 47·11-s + 12·12-s + 35·13-s − 2·14-s − 15·15-s + 16·16-s + 55·17-s − 18·18-s − 125·19-s − 20·20-s + 3·21-s + 94·22-s + 213·23-s − 24·24-s + 25·25-s − 70·26-s + 27·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.0539·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.28·11-s + 0.288·12-s + 0.746·13-s − 0.0381·14-s − 0.258·15-s + 1/4·16-s + 0.784·17-s − 0.235·18-s − 1.50·19-s − 0.223·20-s + 0.0311·21-s + 0.910·22-s + 1.93·23-s − 0.204·24-s + 1/5·25-s − 0.528·26-s + 0.192·27-s + 0.0269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 37 | \( 1 + p T \) |
| good | 7 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 47 T + p^{3} T^{2} \) |
| 13 | \( 1 - 35 T + p^{3} T^{2} \) |
| 17 | \( 1 - 55 T + p^{3} T^{2} \) |
| 19 | \( 1 + 125 T + p^{3} T^{2} \) |
| 23 | \( 1 - 213 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 98 T + p^{3} T^{2} \) |
| 41 | \( 1 + 40 T + p^{3} T^{2} \) |
| 43 | \( 1 + 188 T + p^{3} T^{2} \) |
| 47 | \( 1 - 304 T + p^{3} T^{2} \) |
| 53 | \( 1 - 341 T + p^{3} T^{2} \) |
| 59 | \( 1 + 518 T + p^{3} T^{2} \) |
| 61 | \( 1 + 382 T + p^{3} T^{2} \) |
| 67 | \( 1 + 578 T + p^{3} T^{2} \) |
| 71 | \( 1 - 882 T + p^{3} T^{2} \) |
| 73 | \( 1 + 713 T + p^{3} T^{2} \) |
| 79 | \( 1 - 288 T + p^{3} T^{2} \) |
| 83 | \( 1 + 849 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1263 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1756 T + p^{3} T^{2} \) |
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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
−8.820106579355200427029914785612, −8.372546136208316619855041792795, −7.57940303953154126986597038216, −6.85724978682850641504227787602, −5.71912807900745784806551609284, −4.65254839303622597645582772932, −3.43255804732070327599116390933, −2.61318612093453186156253996926, −1.35105572888509436650987716739, 0,
1.35105572888509436650987716739, 2.61318612093453186156253996926, 3.43255804732070327599116390933, 4.65254839303622597645582772932, 5.71912807900745784806551609284, 6.85724978682850641504227787602, 7.57940303953154126986597038216, 8.372546136208316619855041792795, 8.820106579355200427029914785612
