L(s) = 1 | − 3·3-s + 5-s + 6·9-s + 2·11-s − 3·15-s − 4·17-s + 6·19-s + 3·23-s + 25-s − 9·27-s − 9·29-s − 4·31-s − 6·33-s + 4·37-s − 7·41-s + 5·43-s + 6·45-s + 8·47-s + 12·51-s + 2·53-s + 2·55-s − 18·57-s − 10·59-s − 61-s + 9·67-s − 9·69-s + 2·71-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s + 0.603·11-s − 0.774·15-s − 0.970·17-s + 1.37·19-s + 0.625·23-s + 1/5·25-s − 1.73·27-s − 1.67·29-s − 0.718·31-s − 1.04·33-s + 0.657·37-s − 1.09·41-s + 0.762·43-s + 0.894·45-s + 1.16·47-s + 1.68·51-s + 0.274·53-s + 0.269·55-s − 2.38·57-s − 1.30·59-s − 0.128·61-s + 1.09·67-s − 1.08·69-s + 0.237·71-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
\(L(1)\) |
\(\approx\) |
\(1.161019715\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.161019715\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.10313281490497, −15.61831776293253, −15.01526956190044, −14.35231221015476, −13.49389795903986, −13.25827893435333, −12.41480743012073, −12.11341821365255, −11.27916444106050, −11.14322993779387, −10.55773282249377, −9.739231785632779, −9.353654679235816, −8.746367455555247, −7.492623597494513, −7.260079624027713, −6.453830749698008, −6.020994275386899, −5.353481420684285, −4.945901921130256, −4.139356461514142, −3.413288875190246, −2.195119768561681, −1.370163194962922, −0.5580680842606200,
0.5580680842606200, 1.370163194962922, 2.195119768561681, 3.413288875190246, 4.139356461514142, 4.945901921130256, 5.353481420684285, 6.020994275386899, 6.453830749698008, 7.260079624027713, 7.492623597494513, 8.746367455555247, 9.353654679235816, 9.739231785632779, 10.55773282249377, 11.14322993779387, 11.27916444106050, 12.11341821365255, 12.41480743012073, 13.25827893435333, 13.49389795903986, 14.35231221015476, 15.01526956190044, 15.61831776293253, 16.10313281490497