L(s) = 1 | − 3-s − 2·7-s + 9-s + 4.47·11-s − 4.47·13-s + 4.47·17-s + 2·21-s − 4·23-s − 27-s + 4·29-s − 8.94·31-s − 4.47·33-s + 4.47·37-s + 4.47·39-s + 10·41-s + 4·43-s − 8·47-s − 3·49-s − 4.47·51-s + 4.47·53-s − 13.4·59-s + 10·61-s − 2·63-s + 8·67-s + 4·69-s + 8.94·71-s + 8.94·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 0.333·9-s + 1.34·11-s − 1.24·13-s + 1.08·17-s + 0.436·21-s − 0.834·23-s − 0.192·27-s + 0.742·29-s − 1.60·31-s − 0.778·33-s + 0.735·37-s + 0.716·39-s + 1.56·41-s + 0.609·43-s − 1.16·47-s − 0.428·49-s − 0.626·51-s + 0.614·53-s − 1.74·59-s + 1.28·61-s − 0.251·63-s + 0.977·67-s + 0.481·69-s + 1.06·71-s + 1.04·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.260776783\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.260776783\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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
−9.394923561499792875278883090299, −8.054586098693156195597943569590, −7.33687140914472755794897013319, −6.55678835939007687340089477271, −5.95339759999321565514956373542, −5.07083261314643721631974564941, −4.11514933551697702845382619408, −3.32972740334086786409784125902, −2.08109084610112438217265453635, −0.73889573525456845550463854024,
0.73889573525456845550463854024, 2.08109084610112438217265453635, 3.32972740334086786409784125902, 4.11514933551697702845382619408, 5.07083261314643721631974564941, 5.95339759999321565514956373542, 6.55678835939007687340089477271, 7.33687140914472755794897013319, 8.054586098693156195597943569590, 9.394923561499792875278883090299