L(s) = 1 | − 5-s + 4·7-s − 11-s − 13-s − 4·17-s + 2·19-s + 6·23-s + 25-s + 6·29-s + 2·31-s − 4·35-s − 8·37-s − 6·41-s − 4·43-s − 4·47-s + 9·49-s + 55-s − 4·59-s − 2·61-s + 65-s − 8·67-s − 4·77-s + 8·79-s − 4·83-s + 4·85-s + 18·89-s − 4·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.301·11-s − 0.277·13-s − 0.970·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.676·35-s − 1.31·37-s − 0.937·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s + 0.134·55-s − 0.520·59-s − 0.256·61-s + 0.124·65-s − 0.977·67-s − 0.455·77-s + 0.900·79-s − 0.439·83-s + 0.433·85-s + 1.90·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.94230920947030, −13.50028209757201, −13.13173403086741, −12.33857465258979, −11.84586764641684, −11.70616698427695, −10.89765500137963, −10.74872446788324, −10.19800633884421, −9.448966954500040, −8.839705889077947, −8.525543678810674, −7.969013555431400, −7.600700267728323, −6.829617842407213, −6.678471056433935, −5.692528608322429, −5.058224359686257, −4.780705806433343, −4.396540665335473, −3.475075055200349, −2.989526746506650, −2.214513801121699, −1.612442301806406, −0.9489586457681049, 0,
0.9489586457681049, 1.612442301806406, 2.214513801121699, 2.989526746506650, 3.475075055200349, 4.396540665335473, 4.780705806433343, 5.058224359686257, 5.692528608322429, 6.678471056433935, 6.829617842407213, 7.600700267728323, 7.969013555431400, 8.525543678810674, 8.839705889077947, 9.448966954500040, 10.19800633884421, 10.74872446788324, 10.89765500137963, 11.70616698427695, 11.84586764641684, 12.33857465258979, 13.13173403086741, 13.50028209757201, 13.94230920947030