| L(s) = 1 | + 3·3-s + 7-s + 6·9-s − 3·11-s − 13-s + 5·17-s + 8·19-s + 3·21-s + 2·23-s + 9·27-s − 29-s + 2·31-s − 9·33-s − 10·37-s − 3·39-s − 6·41-s − 4·43-s + 11·47-s + 49-s + 15·51-s − 6·53-s + 24·57-s + 10·59-s + 6·63-s − 10·67-s + 6·69-s + 10·73-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s − 0.904·11-s − 0.277·13-s + 1.21·17-s + 1.83·19-s + 0.654·21-s + 0.417·23-s + 1.73·27-s − 0.185·29-s + 0.359·31-s − 1.56·33-s − 1.64·37-s − 0.480·39-s − 0.937·41-s − 0.609·43-s + 1.60·47-s + 1/7·49-s + 2.10·51-s − 0.824·53-s + 3.17·57-s + 1.30·59-s + 0.755·63-s − 1.22·67-s + 0.722·69-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.789709525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.789709525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
| 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
−8.736688182941118664061272291560, −7.977956718238893242201598667426, −7.59170509290778688699017319946, −6.93074631845068003962696666554, −5.45894122025869864240991912809, −4.92564320979575391817123104740, −3.63741975937206848034338064764, −3.16068121046657476391499347230, −2.28807159941308516225763507700, −1.23084531783742094651227376461,
1.23084531783742094651227376461, 2.28807159941308516225763507700, 3.16068121046657476391499347230, 3.63741975937206848034338064764, 4.92564320979575391817123104740, 5.45894122025869864240991912809, 6.93074631845068003962696666554, 7.59170509290778688699017319946, 7.977956718238893242201598667426, 8.736688182941118664061272291560
