| L(s) = 1 | − 3-s − 7-s + 9-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s + 21-s − 8·23-s − 27-s + 2·29-s + 4·33-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s + 49-s + 2·51-s − 10·53-s − 4·57-s + 12·59-s − 14·61-s − 63-s + 12·67-s + 8·69-s + 8·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-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 + 0.696·33-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 − 1.37·53-s − 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.125·63-s + 1.46·67-s + 0.963·69-s + 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5450538133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5450538133\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.05146929040524, −14.48460753439380, −13.84952028200928, −13.35026518218478, −12.94142556981642, −12.16307931110177, −12.03942298125522, −11.24207127239058, −10.76405025156616, −10.15159124919721, −9.771074946824560, −9.281714285008253, −8.363909727651386, −7.864656106252926, −7.442415904472398, −6.663263761881986, −6.194262845704831, −5.485246280804697, −5.082297771447091, −4.371037525497416, −3.722508147679405, −2.821537767358593, −2.339113694142697, −1.379795878406300, −0.2885885084806375,
0.2885885084806375, 1.379795878406300, 2.339113694142697, 2.821537767358593, 3.722508147679405, 4.371037525497416, 5.082297771447091, 5.485246280804697, 6.194262845704831, 6.663263761881986, 7.442415904472398, 7.864656106252926, 8.363909727651386, 9.281714285008253, 9.771074946824560, 10.15159124919721, 10.76405025156616, 11.24207127239058, 12.03942298125522, 12.16307931110177, 12.94142556981642, 13.35026518218478, 13.84952028200928, 14.48460753439380, 15.05146929040524
