L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 3·7-s + 9-s − 5·11-s − 2·12-s − 2·13-s + 6·14-s − 4·16-s + 6·17-s + 2·18-s − 6·19-s − 3·21-s − 10·22-s − 9·23-s − 4·26-s − 27-s + 6·28-s + 29-s − 2·31-s − 8·32-s + 5·33-s + 12·34-s + 2·36-s − 37-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1.13·7-s + 1/3·9-s − 1.50·11-s − 0.577·12-s − 0.554·13-s + 1.60·14-s − 16-s + 1.45·17-s + 0.471·18-s − 1.37·19-s − 0.654·21-s − 2.13·22-s − 1.87·23-s − 0.784·26-s − 0.192·27-s + 1.13·28-s + 0.185·29-s − 0.359·31-s − 1.41·32-s + 0.870·33-s + 2.05·34-s + 1/3·36-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83234803660229109588865904558, −7.54740476343963674549472631818, −6.24357414292211393153596697451, −5.78180778565250228622005721493, −5.02594328209869903685800983902, −4.60956624547012436038730148171, −3.74543273200017646223279827807, −2.64397051882091468672200773297, −1.82835020575181270599988730084, 0,
1.82835020575181270599988730084, 2.64397051882091468672200773297, 3.74543273200017646223279827807, 4.60956624547012436038730148171, 5.02594328209869903685800983902, 5.78180778565250228622005721493, 6.24357414292211393153596697451, 7.54740476343963674549472631818, 7.83234803660229109588865904558