L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 4·7-s + 8-s + 9-s − 2·10-s − 11-s + 12-s − 4·13-s − 4·14-s − 2·15-s + 16-s − 17-s + 18-s − 8·19-s − 2·20-s − 4·21-s − 22-s + 24-s − 25-s − 4·26-s + 27-s − 4·28-s − 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s − 0.447·20-s − 0.872·21-s − 0.213·22-s + 0.204·24-s − 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.755·28-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.560427624413729086554795114299, −8.435750833845311104302153360400, −7.74368678199029879196940975611, −6.74754903973782460909532845280, −6.26511675492952045832350660509, −4.80277085351441108727338834258, −4.06804449982460236237661592805, −3.14298404820235674746724047452, −2.35879996626412509539094201232, 0,
2.35879996626412509539094201232, 3.14298404820235674746724047452, 4.06804449982460236237661592805, 4.80277085351441108727338834258, 6.26511675492952045832350660509, 6.74754903973782460909532845280, 7.74368678199029879196940975611, 8.435750833845311104302153360400, 9.560427624413729086554795114299