| L(s) = 1 | − 0.0639·2-s + 2.10·3-s − 1.99·4-s − 3.38·5-s − 0.134·6-s + 3.96·7-s + 0.255·8-s + 1.41·9-s + 0.216·10-s − 11-s − 4.19·12-s − 7.07·13-s − 0.253·14-s − 7.12·15-s + 3.97·16-s + 17-s − 0.0904·18-s − 2.30·19-s + 6.76·20-s + 8.33·21-s + 0.0639·22-s + 0.134·23-s + 0.537·24-s + 6.48·25-s + 0.452·26-s − 3.33·27-s − 7.92·28-s + ⋯ |
| L(s) = 1 | − 0.0452·2-s + 1.21·3-s − 0.997·4-s − 1.51·5-s − 0.0548·6-s + 1.49·7-s + 0.0903·8-s + 0.471·9-s + 0.0685·10-s − 0.301·11-s − 1.21·12-s − 1.96·13-s − 0.0678·14-s − 1.83·15-s + 0.993·16-s + 0.242·17-s − 0.0213·18-s − 0.528·19-s + 1.51·20-s + 1.81·21-s + 0.0136·22-s + 0.0280·23-s + 0.109·24-s + 1.29·25-s + 0.0887·26-s − 0.641·27-s − 1.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.354511189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.354511189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0639T + 2T^{2} \) |
| 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 - 0.134T + 23T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 - 0.217T + 31T^{2} \) |
| 37 | \( 1 - 0.290T + 37T^{2} \) |
| 41 | \( 1 + 8.30T + 41T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 4.02T + 53T^{2} \) |
| 59 | \( 1 - 8.42T + 59T^{2} \) |
| 61 | \( 1 - 8.90T + 61T^{2} \) |
| 67 | \( 1 - 1.21T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 2.49T + 73T^{2} \) |
| 79 | \( 1 - 7.56T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−7.971020862585337640339139123748, −7.59669004622036346969011225933, −6.84684965392542587883450193728, −5.20042759917304375485953033193, −4.93391099752447333601794725377, −4.25071566758729743662144646788, −3.61201340423691315048910717511, −2.76886239771868924203050048007, −1.90740131038436448465026338698, −0.53691700511491279626545975936,
0.53691700511491279626545975936, 1.90740131038436448465026338698, 2.76886239771868924203050048007, 3.61201340423691315048910717511, 4.25071566758729743662144646788, 4.93391099752447333601794725377, 5.20042759917304375485953033193, 6.84684965392542587883450193728, 7.59669004622036346969011225933, 7.971020862585337640339139123748
