L(s) = 1 | − 2·3-s − 5·9-s − 14·11-s − 2·17-s − 34·19-s + 25·25-s + 28·27-s + 28·33-s + 46·41-s − 14·43-s + 4·51-s + 68·57-s − 82·59-s − 62·67-s + 142·73-s − 50·75-s − 11·81-s + 158·83-s − 146·89-s + 94·97-s + 70·99-s + 178·107-s + 98·113-s + ⋯ |
L(s) = 1 | − 2/3·3-s − 5/9·9-s − 1.27·11-s − 0.117·17-s − 1.78·19-s + 25-s + 1.03·27-s + 0.848·33-s + 1.12·41-s − 0.325·43-s + 4/51·51-s + 1.19·57-s − 1.38·59-s − 0.925·67-s + 1.94·73-s − 2/3·75-s − 0.135·81-s + 1.90·83-s − 1.64·89-s + 0.969·97-s + 0.707·99-s + 1.66·107-s + 0.867·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8273430433\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8273430433\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p^{2} T^{2} \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 + 14 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 + 2 T + p^{2} T^{2} \) |
| 19 | \( 1 + 34 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 - 46 T + p^{2} T^{2} \) |
| 43 | \( 1 + 14 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( 1 + 82 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 + 62 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 142 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 - 158 T + p^{2} T^{2} \) |
| 89 | \( 1 + 146 T + p^{2} T^{2} \) |
| 97 | \( 1 - 94 T + p^{2} T^{2} \) |
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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
−9.152395981110827228439684658366, −8.435746453629335028398027984578, −7.71120773380786649992907513798, −6.64026491016849692559864570405, −6.00070060644862675561151852794, −5.13382945808920307768567833954, −4.43050146077250669005907803669, −3.07989435337194361346381975256, −2.17318398502561784555681740847, −0.50274953242930439082148416241,
0.50274953242930439082148416241, 2.17318398502561784555681740847, 3.07989435337194361346381975256, 4.43050146077250669005907803669, 5.13382945808920307768567833954, 6.00070060644862675561151852794, 6.64026491016849692559864570405, 7.71120773380786649992907513798, 8.435746453629335028398027984578, 9.152395981110827228439684658366