| L(s) = 1 | − 3·3-s − 16·5-s − 7·7-s + 9·9-s + 18·11-s − 54·13-s + 48·15-s − 128·17-s − 52·19-s + 21·21-s + 202·23-s + 131·25-s − 27·27-s + 302·29-s + 200·31-s − 54·33-s + 112·35-s − 150·37-s + 162·39-s + 172·41-s − 164·43-s − 144·45-s + 460·47-s + 49·49-s + 384·51-s − 190·53-s − 288·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.43·5-s − 0.377·7-s + 1/3·9-s + 0.493·11-s − 1.15·13-s + 0.826·15-s − 1.82·17-s − 0.627·19-s + 0.218·21-s + 1.83·23-s + 1.04·25-s − 0.192·27-s + 1.93·29-s + 1.15·31-s − 0.284·33-s + 0.540·35-s − 0.666·37-s + 0.665·39-s + 0.655·41-s − 0.581·43-s − 0.477·45-s + 1.42·47-s + 1/7·49-s + 1.05·51-s − 0.492·53-s − 0.706·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7409806101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7409806101\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 128 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 202 T + p^{3} T^{2} \) |
| 29 | \( 1 - 302 T + p^{3} T^{2} \) |
| 31 | \( 1 - 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 150 T + p^{3} T^{2} \) |
| 41 | \( 1 - 172 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 460 T + p^{3} T^{2} \) |
| 53 | \( 1 + 190 T + p^{3} T^{2} \) |
| 59 | \( 1 + 96 T + p^{3} T^{2} \) |
| 61 | \( 1 - 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 744 T + p^{3} T^{2} \) |
| 71 | \( 1 - 54 T + p^{3} T^{2} \) |
| 73 | \( 1 - 742 T + p^{3} T^{2} \) |
| 79 | \( 1 - 92 T + p^{3} T^{2} \) |
| 83 | \( 1 - 228 T + p^{3} T^{2} \) |
| 89 | \( 1 + 116 T + p^{3} T^{2} \) |
| 97 | \( 1 + 554 T + p^{3} T^{2} \) |
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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
−11.20876583832361548647321631046, −10.45478755097439234300758133236, −9.173161647440508266531533252313, −8.303044130929428967487671027956, −7.05245144103255058414809493545, −6.59900792753048558806810860192, −4.84938853627883719688575040971, −4.22339317529329783087355723542, −2.74849238853855545214653975714, −0.58171754790662278082856459855,
0.58171754790662278082856459855, 2.74849238853855545214653975714, 4.22339317529329783087355723542, 4.84938853627883719688575040971, 6.59900792753048558806810860192, 7.05245144103255058414809493545, 8.303044130929428967487671027956, 9.173161647440508266531533252313, 10.45478755097439234300758133236, 11.20876583832361548647321631046
