| L(s) = 1 | − 5-s − 3·9-s − 2·11-s + 6·13-s + 25-s + 6·29-s + 6·31-s + 2·37-s − 6·41-s + 6·43-s + 3·45-s + 10·47-s − 6·53-s + 2·55-s + 10·61-s − 6·65-s − 2·67-s − 6·71-s + 6·73-s − 6·79-s + 9·81-s − 6·83-s + 18·89-s − 14·97-s + 6·99-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 0.603·11-s + 1.66·13-s + 1/5·25-s + 1.11·29-s + 1.07·31-s + 0.328·37-s − 0.937·41-s + 0.914·43-s + 0.447·45-s + 1.45·47-s − 0.824·53-s + 0.269·55-s + 1.28·61-s − 0.744·65-s − 0.244·67-s − 0.712·71-s + 0.702·73-s − 0.675·79-s + 81-s − 0.658·83-s + 1.90·89-s − 1.42·97-s + 0.603·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.01723325423618, −12.43596629801200, −11.99252549460012, −11.59347970376099, −11.08560081542624, −10.80456945763315, −10.29560671364010, −9.840670705127775, −9.077898314632108, −8.718626142723249, −8.350195077912490, −8.000328699309774, −7.450122303317294, −6.807991256050961, −6.264748441720430, −5.967908828040805, −5.414943339206476, −4.839302586583543, −4.315547770529531, −3.708063994837242, −3.266571986977518, −2.685678644822408, −2.252073151310180, −1.235772044277751, −0.8302457381477496, 0,
0.8302457381477496, 1.235772044277751, 2.252073151310180, 2.685678644822408, 3.266571986977518, 3.708063994837242, 4.315547770529531, 4.839302586583543, 5.414943339206476, 5.967908828040805, 6.264748441720430, 6.807991256050961, 7.450122303317294, 8.000328699309774, 8.350195077912490, 8.718626142723249, 9.077898314632108, 9.840670705127775, 10.29560671364010, 10.80456945763315, 11.08560081542624, 11.59347970376099, 11.99252549460012, 12.43596629801200, 13.01723325423618
