| L(s) = 1 | + 5-s − 4·11-s − 2·13-s − 2·19-s + 4·23-s + 25-s − 10·29-s − 4·31-s − 2·37-s − 12·41-s − 4·43-s + 4·47-s − 2·53-s − 4·55-s + 10·59-s − 6·61-s − 2·65-s + 4·67-s + 12·71-s + 4·73-s − 4·79-s + 14·83-s + 8·89-s − 2·95-s + 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 0.554·13-s − 0.458·19-s + 0.834·23-s + 1/5·25-s − 1.85·29-s − 0.718·31-s − 0.328·37-s − 1.87·41-s − 0.609·43-s + 0.583·47-s − 0.274·53-s − 0.539·55-s + 1.30·59-s − 0.768·61-s − 0.248·65-s + 0.488·67-s + 1.42·71-s + 0.468·73-s − 0.450·79-s + 1.53·83-s + 0.847·89-s − 0.205·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358361069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358361069\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.69371317312706, −15.24159819541230, −14.84595253505502, −14.18878221577148, −13.52662893701711, −12.98990281910910, −12.78407873635501, −11.94628176755315, −11.33180959992899, −10.68863034494654, −10.32346719971337, −9.621650177270095, −9.115182615768786, −8.443161697443854, −7.800907147218975, −7.207769486406791, −6.665823412300649, −5.819616827348174, −5.170041520445052, −4.920016118148328, −3.789590667211128, −3.194563847594484, −2.267634993274780, −1.810038773775769, −0.4669277381491782,
0.4669277381491782, 1.810038773775769, 2.267634993274780, 3.194563847594484, 3.789590667211128, 4.920016118148328, 5.170041520445052, 5.819616827348174, 6.665823412300649, 7.207769486406791, 7.800907147218975, 8.443161697443854, 9.115182615768786, 9.621650177270095, 10.32346719971337, 10.68863034494654, 11.33180959992899, 11.94628176755315, 12.78407873635501, 12.98990281910910, 13.52662893701711, 14.18878221577148, 14.84595253505502, 15.24159819541230, 15.69371317312706
