L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 6·13-s + 14-s − 15-s + 16-s + 18-s − 2·19-s − 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 6·26-s + 27-s + 28-s + 3·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.978077723\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.978077723\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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
−13.97432237444445, −13.28159861151931, −12.85435640370748, −12.28645503813699, −12.02039334597485, −11.55552229209952, −10.74137089877820, −10.55511297239024, −9.860071167233027, −9.241939249974292, −8.878602663975204, −8.183386849628594, −7.624151914932408, −7.282748756444238, −6.869740378972423, −6.063162077326353, −5.561515198459252, −4.833613544943067, −4.395798116535429, −4.086239557576763, −3.161062947759692, −2.782683290368055, −2.145459950547010, −1.501882409924877, −0.5180304165817774,
0.5180304165817774, 1.501882409924877, 2.145459950547010, 2.782683290368055, 3.161062947759692, 4.086239557576763, 4.395798116535429, 4.833613544943067, 5.561515198459252, 6.063162077326353, 6.869740378972423, 7.282748756444238, 7.624151914932408, 8.183386849628594, 8.878602663975204, 9.241939249974292, 9.860071167233027, 10.55511297239024, 10.74137089877820, 11.55552229209952, 12.02039334597485, 12.28645503813699, 12.85435640370748, 13.28159861151931, 13.97432237444445