| L(s) = 1 | − 2-s + 4-s + 4.22·7-s − 8-s + 5.13·11-s + 3.16·13-s − 4.22·14-s + 16-s + 6.48·17-s − 19-s − 5.13·22-s − 7.56·23-s − 3.16·26-s + 4.22·28-s − 0.832·29-s − 4.51·31-s − 32-s − 6.48·34-s + 0.137·37-s + 38-s + 11.6·41-s − 2.51·43-s + 5.13·44-s + 7.56·46-s + 5.96·47-s + 10.8·49-s + 3.16·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.59·7-s − 0.353·8-s + 1.54·11-s + 0.878·13-s − 1.12·14-s + 0.250·16-s + 1.57·17-s − 0.229·19-s − 1.09·22-s − 1.57·23-s − 0.621·26-s + 0.798·28-s − 0.154·29-s − 0.810·31-s − 0.176·32-s − 1.11·34-s + 0.0226·37-s + 0.162·38-s + 1.81·41-s − 0.382·43-s + 0.774·44-s + 1.11·46-s + 0.869·47-s + 1.55·49-s + 0.439·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.415979273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.415979273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 23 | \( 1 + 7.56T + 23T^{2} \) |
| 29 | \( 1 + 0.832T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 0.137T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 + 0.225T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 + 3.82T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
| 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
−7.86007257298849773419864724099, −7.38474994919853819248159158271, −6.37004817788632636368053731772, −5.86434721722882601978443053771, −5.10229104502581608875967570252, −4.00202739644310170640619634508, −3.71802031671785248718146147970, −2.30525392428765716410928896871, −1.51309853172338444837242952043, −0.970641260981495929752070411776,
0.970641260981495929752070411776, 1.51309853172338444837242952043, 2.30525392428765716410928896871, 3.71802031671785248718146147970, 4.00202739644310170640619634508, 5.10229104502581608875967570252, 5.86434721722882601978443053771, 6.37004817788632636368053731772, 7.38474994919853819248159158271, 7.86007257298849773419864724099
