L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 6·11-s − 12-s + 16-s + 7·17-s + 18-s − 19-s + 6·22-s + 6·23-s − 24-s − 27-s − 4·29-s + 8·31-s + 32-s − 6·33-s + 7·34-s + 36-s + 8·37-s − 38-s − 10·41-s − 4·43-s + 6·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s + 1/4·16-s + 1.69·17-s + 0.235·18-s − 0.229·19-s + 1.27·22-s + 1.25·23-s − 0.204·24-s − 0.192·27-s − 0.742·29-s + 1.43·31-s + 0.176·32-s − 1.04·33-s + 1.20·34-s + 1/6·36-s + 1.31·37-s − 0.162·38-s − 1.56·41-s − 0.609·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.379128596\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.379128596\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.24201079889612, −13.10266887949457, −12.24570042489373, −11.98221331448762, −11.63162029526996, −11.32246246875700, −10.52324365593308, −10.10191958134812, −9.749854911711931, −8.936444016830404, −8.732397098095685, −7.833535395008192, −7.396090499326210, −6.876442110483498, −6.368215795410473, −5.977068611880498, −5.467611854647063, −4.795845372175042, −4.430878430899003, −3.732703840541345, −3.333508130709373, −2.724706422864139, −1.742391026177852, −1.239702592185978, −0.7079398798325962,
0.7079398798325962, 1.239702592185978, 1.742391026177852, 2.724706422864139, 3.333508130709373, 3.732703840541345, 4.430878430899003, 4.795845372175042, 5.467611854647063, 5.977068611880498, 6.368215795410473, 6.876442110483498, 7.396090499326210, 7.833535395008192, 8.732397098095685, 8.936444016830404, 9.749854911711931, 10.10191958134812, 10.52324365593308, 11.32246246875700, 11.63162029526996, 11.98221331448762, 12.24570042489373, 13.10266887949457, 13.24201079889612