L(s) = 1 | + 3-s + 5-s − 2·9-s + 3·11-s + 13-s + 15-s + 5·17-s + 6·19-s + 25-s − 5·27-s + 5·29-s + 2·31-s + 3·33-s + 4·37-s + 39-s + 2·41-s + 10·43-s − 2·45-s − 9·47-s + 5·51-s − 6·53-s + 3·55-s + 6·57-s + 6·59-s − 12·61-s + 65-s − 2·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s + 1.21·17-s + 1.37·19-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 0.359·31-s + 0.522·33-s + 0.657·37-s + 0.160·39-s + 0.312·41-s + 1.52·43-s − 0.298·45-s − 1.31·47-s + 0.700·51-s − 0.824·53-s + 0.404·55-s + 0.794·57-s + 0.781·59-s − 1.53·61-s + 0.124·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.606479563\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.606479563\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−16.09553165773564, −15.37030590430330, −14.66413279455587, −14.28492155828444, −13.90304541396776, −13.47373785213731, −12.52551472005807, −12.19134846961602, −11.41683389736817, −11.10367077432486, −10.09374281395054, −9.693570637309033, −9.183109547057622, −8.602212143766510, −7.898917160759899, −7.491991044823729, −6.554993224172288, −6.014349580511757, −5.443071559191419, −4.680668650596809, −3.779284693238222, −3.151279339538424, −2.628126292269647, −1.540760072985058, −0.8601914177181326,
0.8601914177181326, 1.540760072985058, 2.628126292269647, 3.151279339538424, 3.779284693238222, 4.680668650596809, 5.443071559191419, 6.014349580511757, 6.554993224172288, 7.491991044823729, 7.898917160759899, 8.602212143766510, 9.183109547057622, 9.693570637309033, 10.09374281395054, 11.10367077432486, 11.41683389736817, 12.19134846961602, 12.52551472005807, 13.47373785213731, 13.90304541396776, 14.28492155828444, 14.66413279455587, 15.37030590430330, 16.09553165773564