L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 5·11-s + 3·13-s + 4·14-s + 16-s − 17-s − 6·19-s − 5·22-s − 23-s − 3·26-s − 4·28-s + 9·29-s − 5·31-s − 32-s + 34-s + 2·37-s + 6·38-s + 2·41-s − 43-s + 5·44-s + 46-s + 13·47-s + 9·49-s + 3·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 1.50·11-s + 0.832·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s − 1.06·22-s − 0.208·23-s − 0.588·26-s − 0.755·28-s + 1.67·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s + 0.328·37-s + 0.973·38-s + 0.312·41-s − 0.152·43-s + 0.753·44-s + 0.147·46-s + 1.89·47-s + 9/7·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.023017702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.023017702\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.432701532400887850792686091088, −8.958692508219409519666615479852, −8.238439826174504573338277595221, −6.89866891087402561240332449158, −6.53598316211219128918852273783, −5.86814811044628806480862801402, −4.19485731029531005053102055212, −3.49678776549837799496194532787, −2.27585805022063955745898798949, −0.817452243984419354548042334758,
0.817452243984419354548042334758, 2.27585805022063955745898798949, 3.49678776549837799496194532787, 4.19485731029531005053102055212, 5.86814811044628806480862801402, 6.53598316211219128918852273783, 6.89866891087402561240332449158, 8.238439826174504573338277595221, 8.958692508219409519666615479852, 9.432701532400887850792686091088