L(s) = 1 | − 3·3-s − 5-s − 5·7-s + 6·9-s − 2·11-s + 4·13-s + 3·15-s + 17-s − 3·19-s + 15·21-s − 4·23-s − 4·25-s − 9·27-s − 3·29-s + 8·31-s + 6·33-s + 5·35-s − 12·39-s + 3·41-s + 4·43-s − 6·45-s + 8·47-s + 18·49-s − 3·51-s + 7·53-s + 2·55-s + 9·57-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.447·5-s − 1.88·7-s + 2·9-s − 0.603·11-s + 1.10·13-s + 0.774·15-s + 0.242·17-s − 0.688·19-s + 3.27·21-s − 0.834·23-s − 4/5·25-s − 1.73·27-s − 0.557·29-s + 1.43·31-s + 1.04·33-s + 0.845·35-s − 1.92·39-s + 0.468·41-s + 0.609·43-s − 0.894·45-s + 1.16·47-s + 18/7·49-s − 0.420·51-s + 0.961·53-s + 0.269·55-s + 1.19·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.903300837174390911035528850888, −7.08614803891019696892062331006, −6.32903379197905050458300817698, −6.03188896603336529912461647418, −5.36925875781838972355737971399, −4.14700144449420769146954143753, −3.75458992923721340994900991768, −2.49870997660020956877328607924, −0.865766667653795111763984925961, 0,
0.865766667653795111763984925961, 2.49870997660020956877328607924, 3.75458992923721340994900991768, 4.14700144449420769146954143753, 5.36925875781838972355737971399, 6.03188896603336529912461647418, 6.32903379197905050458300817698, 7.08614803891019696892062331006, 7.903300837174390911035528850888