| L(s) = 1 | − 3·9-s − 4·11-s + 13-s − 6·17-s − 6·19-s − 23-s − 5·25-s + 10·29-s − 2·31-s + 10·37-s − 8·41-s − 12·43-s + 10·47-s + 2·53-s − 8·59-s − 8·61-s − 4·67-s + 8·71-s − 16·73-s + 8·79-s + 9·81-s + 6·83-s − 2·89-s − 10·97-s + 12·99-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 9-s − 1.20·11-s + 0.277·13-s − 1.45·17-s − 1.37·19-s − 0.208·23-s − 25-s + 1.85·29-s − 0.359·31-s + 1.64·37-s − 1.24·41-s − 1.82·43-s + 1.45·47-s + 0.274·53-s − 1.04·59-s − 1.02·61-s − 0.488·67-s + 0.949·71-s − 1.87·73-s + 0.900·79-s + 81-s + 0.658·83-s − 0.211·89-s − 1.01·97-s + 1.20·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.33734180317843, −12.70311568609746, −12.09017452748538, −11.87902858838293, −11.14324221814391, −10.82510688925310, −10.51819059889277, −9.944268857081947, −9.385359583862075, −8.791641345323358, −8.363757701026971, −8.177935150526146, −7.588970231844389, −6.815821602580326, −6.480188204610440, −5.949528423316029, −5.568920129667845, −4.775778049551362, −4.527619318241575, −3.926142364507142, −3.112981223989513, −2.705030041083820, −2.199302653148780, −1.646969011448375, −0.5230780119192186, 0,
0.5230780119192186, 1.646969011448375, 2.199302653148780, 2.705030041083820, 3.112981223989513, 3.926142364507142, 4.527619318241575, 4.775778049551362, 5.568920129667845, 5.949528423316029, 6.480188204610440, 6.815821602580326, 7.588970231844389, 8.177935150526146, 8.363757701026971, 8.791641345323358, 9.385359583862075, 9.944268857081947, 10.51819059889277, 10.82510688925310, 11.14324221814391, 11.87902858838293, 12.09017452748538, 12.70311568609746, 13.33734180317843
