L(s) = 1 | − 3·7-s − 3·11-s + 13-s − 17-s + 6·19-s + 5·23-s + 6·29-s − 2·31-s − 7·37-s − 3·41-s − 8·43-s + 2·47-s + 2·49-s − 53-s − 15·61-s + 12·67-s + 5·71-s + 6·73-s + 9·77-s + 13·79-s − 12·83-s − 89-s − 3·91-s − 17·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 1.13·7-s − 0.904·11-s + 0.277·13-s − 0.242·17-s + 1.37·19-s + 1.04·23-s + 1.11·29-s − 0.359·31-s − 1.15·37-s − 0.468·41-s − 1.21·43-s + 0.291·47-s + 2/7·49-s − 0.137·53-s − 1.92·61-s + 1.46·67-s + 0.593·71-s + 0.702·73-s + 1.02·77-s + 1.46·79-s − 1.31·83-s − 0.105·89-s − 0.314·91-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−15.09273121797374, −14.10943347680783, −13.78661486308755, −13.36448769280820, −12.80445091409550, −12.34115516516754, −11.89786891691030, −11.07554164171063, −10.78999117829661, −10.02493324480397, −9.759693927847167, −9.143133588799636, −8.569352708522115, −8.007877058325210, −7.357966451608734, −6.758417879921319, −6.470302300554495, −5.542808305083119, −5.208570064363647, −4.573812190155509, −3.602488045361523, −3.174481431604218, −2.728094441908770, −1.760454023817664, −0.8649557052162369, 0,
0.8649557052162369, 1.760454023817664, 2.728094441908770, 3.174481431604218, 3.602488045361523, 4.573812190155509, 5.208570064363647, 5.542808305083119, 6.470302300554495, 6.758417879921319, 7.357966451608734, 8.007877058325210, 8.569352708522115, 9.143133588799636, 9.759693927847167, 10.02493324480397, 10.78999117829661, 11.07554164171063, 11.89786891691030, 12.34115516516754, 12.80445091409550, 13.36448769280820, 13.78661486308755, 14.10943347680783, 15.09273121797374