| L(s) = 1 | − 3-s + 7-s + 9-s − 2·13-s + 2·17-s + 4·19-s − 21-s − 8·23-s − 27-s − 2·29-s + 4·31-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s + 49-s − 2·51-s − 6·53-s − 4·57-s + 6·61-s + 63-s − 4·67-s + 8·69-s + 8·71-s − 10·73-s + 12·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.218·21-s − 1.66·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.768·61-s + 0.125·63-s − 0.488·67-s + 0.963·69-s + 0.949·71-s − 1.17·73-s + 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.566907421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566907421\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.83523270779438, −15.55557668384248, −14.65802126433458, −14.34815103586327, −13.68088156019912, −13.16890651951445, −12.28828021404275, −12.09134776250032, −11.46860446378346, −10.97403968895393, −10.09615758000710, −9.901112413988677, −9.221220412556150, −8.328451846091253, −7.804823025588084, −7.338515673915603, −6.524268687244111, −5.911913383986480, −5.352332983434841, −4.691320567363117, −4.052047442011862, −3.242651589420239, −2.339388509266725, −1.526440599446760, −0.5668825234417538,
0.5668825234417538, 1.526440599446760, 2.339388509266725, 3.242651589420239, 4.052047442011862, 4.691320567363117, 5.352332983434841, 5.911913383986480, 6.524268687244111, 7.338515673915603, 7.804823025588084, 8.328451846091253, 9.221220412556150, 9.901112413988677, 10.09615758000710, 10.97403968895393, 11.46860446378346, 12.09134776250032, 12.28828021404275, 13.16890651951445, 13.68088156019912, 14.34815103586327, 14.65802126433458, 15.55557668384248, 15.83523270779438
