L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s − 2·11-s + 12-s + 6·13-s − 2·14-s + 16-s + 4·17-s − 18-s + 2·21-s + 2·22-s − 23-s − 24-s − 6·26-s + 27-s + 2·28-s + 2·29-s − 32-s − 2·33-s − 4·34-s + 36-s + 8·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.436·21-s + 0.426·22-s − 0.208·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s + 0.377·28-s + 0.371·29-s − 0.176·32-s − 0.348·33-s − 0.685·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.066871698\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.066871698\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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
−8.309963158122306622621312099145, −8.163558217089928800086894582342, −7.42594425912372754419505316886, −6.42666711232414110232916679481, −5.72475094840184530334582513380, −4.75166838037863343829025138132, −3.72547723100288581279665503428, −2.93742558018347709867015445737, −1.83998975193400344249553971739, −0.989040409199339694470251310139,
0.989040409199339694470251310139, 1.83998975193400344249553971739, 2.93742558018347709867015445737, 3.72547723100288581279665503428, 4.75166838037863343829025138132, 5.72475094840184530334582513380, 6.42666711232414110232916679481, 7.42594425912372754419505316886, 8.163558217089928800086894582342, 8.309963158122306622621312099145