| L(s) = 1 | − 5-s + 2·11-s − 2·13-s + 4·17-s − 19-s − 23-s − 4·25-s + 9·29-s + 2·31-s + 4·43-s + 9·47-s − 7·49-s + 5·53-s − 2·55-s − 10·59-s + 4·61-s + 2·65-s + 11·67-s + 13·71-s + 73-s + 4·79-s + 16·83-s − 4·85-s + 95-s − 13·97-s + 9·101-s + 16·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s − 0.554·13-s + 0.970·17-s − 0.229·19-s − 0.208·23-s − 4/5·25-s + 1.67·29-s + 0.359·31-s + 0.609·43-s + 1.31·47-s − 49-s + 0.686·53-s − 0.269·55-s − 1.30·59-s + 0.512·61-s + 0.248·65-s + 1.34·67-s + 1.54·71-s + 0.117·73-s + 0.450·79-s + 1.75·83-s − 0.433·85-s + 0.102·95-s − 1.31·97-s + 0.895·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.600336372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600336372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.246742262283024171659882300933, −8.269004255342038240252151056845, −7.74224073848568231802696823119, −6.83469055357111835096996945200, −6.07695106144184215424835306202, −5.09284229041608836191074192342, −4.22173882863450945705435720322, −3.38334097419425704786660234031, −2.26996522341636254549834848831, −0.862900137723047724558311026793,
0.862900137723047724558311026793, 2.26996522341636254549834848831, 3.38334097419425704786660234031, 4.22173882863450945705435720322, 5.09284229041608836191074192342, 6.07695106144184215424835306202, 6.83469055357111835096996945200, 7.74224073848568231802696823119, 8.269004255342038240252151056845, 9.246742262283024171659882300933
