| L(s) = 1 | + 5-s + 7-s − 5·11-s + 13-s − 3·17-s + 6·19-s − 6·23-s + 25-s + 9·29-s + 35-s + 6·37-s − 8·41-s − 6·43-s + 3·47-s + 49-s + 12·53-s − 5·55-s + 8·59-s − 4·61-s + 65-s + 4·67-s + 8·71-s + 10·73-s − 5·77-s + 3·79-s − 12·83-s − 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.50·11-s + 0.277·13-s − 0.727·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.169·35-s + 0.986·37-s − 1.24·41-s − 0.914·43-s + 0.437·47-s + 1/7·49-s + 1.64·53-s − 0.674·55-s + 1.04·59-s − 0.512·61-s + 0.124·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.569·77-s + 0.337·79-s − 1.31·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.979681115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979681115\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.221402281878789465089589313507, −7.62954374641588235718006492065, −6.79230502198732864314633745798, −6.01444826736210837468924519845, −5.24154681791164573770536092994, −4.76706705720314731476536835342, −3.68523744465265841556690270503, −2.71239320609283552284776322612, −2.03744261910962500613971262945, −0.76044480432284251382048398807,
0.76044480432284251382048398807, 2.03744261910962500613971262945, 2.71239320609283552284776322612, 3.68523744465265841556690270503, 4.76706705720314731476536835342, 5.24154681791164573770536092994, 6.01444826736210837468924519845, 6.79230502198732864314633745798, 7.62954374641588235718006492065, 8.221402281878789465089589313507
