| L(s) = 1 | − 2·7-s − 2·11-s + 13-s − 2·17-s + 2·19-s − 8·23-s + 6·29-s − 2·31-s − 2·37-s + 2·41-s + 6·47-s − 3·49-s − 10·53-s + 14·59-s − 10·61-s − 2·67-s + 6·71-s − 2·73-s + 4·77-s − 12·79-s + 6·83-s + 18·89-s − 2·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.603·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s − 1.66·23-s + 1.11·29-s − 0.359·31-s − 0.328·37-s + 0.312·41-s + 0.875·47-s − 3/7·49-s − 1.37·53-s + 1.82·59-s − 1.28·61-s − 0.244·67-s + 0.712·71-s − 0.234·73-s + 0.455·77-s − 1.35·79-s + 0.658·83-s + 1.90·89-s − 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.011215219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.011215219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.88288399159644, −13.26015103604171, −12.90529461226057, −12.39982380861492, −11.85076453722694, −11.48535938364979, −10.71825311332142, −10.35287292040687, −9.937222853996865, −9.357502888705770, −8.883880299734007, −8.280717254858663, −7.786954346106949, −7.323920103677969, −6.565228534025015, −6.237246035292447, −5.714878889012075, −5.032189639664238, −4.496051093839689, −3.791492725993128, −3.338389050679987, −2.610490783757927, −2.106351958931831, −1.240177114874315, −0.3265307516069062,
0.3265307516069062, 1.240177114874315, 2.106351958931831, 2.610490783757927, 3.338389050679987, 3.791492725993128, 4.496051093839689, 5.032189639664238, 5.714878889012075, 6.237246035292447, 6.565228534025015, 7.323920103677969, 7.786954346106949, 8.280717254858663, 8.883880299734007, 9.357502888705770, 9.937222853996865, 10.35287292040687, 10.71825311332142, 11.48535938364979, 11.85076453722694, 12.39982380861492, 12.90529461226057, 13.26015103604171, 13.88288399159644
