| L(s) = 1 | − 2·5-s + 2·7-s + 4·11-s − 13-s + 6·19-s − 4·23-s − 25-s − 8·29-s + 2·31-s − 4·35-s + 6·37-s + 6·41-s + 8·43-s − 8·47-s − 3·49-s + 12·53-s − 8·55-s − 4·59-s + 10·61-s + 2·65-s + 2·67-s + 16·71-s + 14·73-s + 8·77-s + 4·79-s + 12·83-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.20·11-s − 0.277·13-s + 1.37·19-s − 0.834·23-s − 1/5·25-s − 1.48·29-s + 0.359·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 1.64·53-s − 1.07·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s + 0.244·67-s + 1.89·71-s + 1.63·73-s + 0.911·77-s + 0.450·79-s + 1.31·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693397111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693397111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.360289479379964628298340335373, −8.241852054238896003751921688443, −7.74744660409673229233588960971, −7.04653430603132179733791135396, −6.01414271153323865134271941039, −5.11879297445387160021261131252, −4.12069288502841648092256676316, −3.60367057667107139122670650923, −2.17613657202902066097003258830, −0.916959205246300730998922019849,
0.916959205246300730998922019849, 2.17613657202902066097003258830, 3.60367057667107139122670650923, 4.12069288502841648092256676316, 5.11879297445387160021261131252, 6.01414271153323865134271941039, 7.04653430603132179733791135396, 7.74744660409673229233588960971, 8.241852054238896003751921688443, 9.360289479379964628298340335373
