L(s) = 1 | − 3-s − 4·7-s + 9-s − 13-s − 17-s + 4·19-s + 4·21-s + 8·23-s − 27-s − 10·29-s − 4·31-s − 10·37-s + 39-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s + 51-s + 10·53-s − 4·57-s + 4·59-s − 2·61-s − 4·63-s + 12·67-s − 8·69-s − 12·71-s + 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.872·21-s + 1.66·23-s − 0.192·27-s − 1.85·29-s − 0.718·31-s − 1.64·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.140·51-s + 1.37·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.503·63-s + 1.46·67-s − 0.963·69-s − 1.42·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.50292723548558, −13.11013875086212, −12.81795370634520, −12.36057440914520, −11.72273980680137, −11.37267383575638, −10.83839049530163, −10.24320343332868, −9.903781315984746, −9.361233164876288, −8.978014419022073, −8.505736669082043, −7.505906034094400, −7.218382770734016, −6.805426179364610, −6.355816845320284, −5.532183102658995, −5.378666803202544, −4.779627580326735, −3.877595068488835, −3.436363749807593, −3.082463476505090, −2.206701104842471, −1.530012630780078, −0.6354386577313079, 0,
0.6354386577313079, 1.530012630780078, 2.206701104842471, 3.082463476505090, 3.436363749807593, 3.877595068488835, 4.779627580326735, 5.378666803202544, 5.532183102658995, 6.355816845320284, 6.805426179364610, 7.218382770734016, 7.505906034094400, 8.505736669082043, 8.978014419022073, 9.361233164876288, 9.903781315984746, 10.24320343332868, 10.83839049530163, 11.37267383575638, 11.72273980680137, 12.36057440914520, 12.81795370634520, 13.11013875086212, 13.50292723548558