| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 4·11-s − 6·13-s + 15-s − 2·17-s + 8·19-s − 21-s − 4·23-s + 25-s − 27-s + 6·29-s − 4·31-s − 4·33-s − 35-s − 2·37-s + 6·39-s − 2·41-s + 12·43-s − 45-s + 49-s + 2·51-s + 2·53-s − 4·55-s − 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.258·15-s − 0.485·17-s + 1.83·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.696·33-s − 0.169·35-s − 0.328·37-s + 0.960·39-s − 0.312·41-s + 1.82·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.539·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.301579302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301579302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.489822071500421838123299377246, −8.589885155018342735645511409941, −7.48194518995166404806621976813, −7.15438604252358823564137535690, −6.11635555330347224666598894965, −5.15730774215430543096989146351, −4.48701468714906907010956142437, −3.52333401998483593075569376220, −2.20468163292461647890581529227, −0.825713056820135641492552995973,
0.825713056820135641492552995973, 2.20468163292461647890581529227, 3.52333401998483593075569376220, 4.48701468714906907010956142437, 5.15730774215430543096989146351, 6.11635555330347224666598894965, 7.15438604252358823564137535690, 7.48194518995166404806621976813, 8.589885155018342735645511409941, 9.489822071500421838123299377246
