| L(s) = 1 | − 3-s − 3.41·5-s + 0.585·7-s + 9-s − 2·11-s + 2.82·13-s + 3.41·15-s − 7.65·17-s − 5.65·19-s − 0.585·21-s + 6.82·23-s + 6.65·25-s − 27-s − 3.41·29-s + 7.41·31-s + 2·33-s − 2·35-s − 1.65·37-s − 2.82·39-s − 0.343·41-s + 9.65·43-s − 3.41·45-s − 4.48·47-s − 6.65·49-s + 7.65·51-s + 7.89·53-s + 6.82·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.52·5-s + 0.221·7-s + 0.333·9-s − 0.603·11-s + 0.784·13-s + 0.881·15-s − 1.85·17-s − 1.29·19-s − 0.127·21-s + 1.42·23-s + 1.33·25-s − 0.192·27-s − 0.634·29-s + 1.33·31-s + 0.348·33-s − 0.338·35-s − 0.272·37-s − 0.452·39-s − 0.0535·41-s + 1.47·43-s − 0.508·45-s − 0.654·47-s − 0.950·49-s + 1.07·51-s + 1.08·53-s + 0.920·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7542152054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7542152054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 0.343T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 7.89T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 9.65T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−9.307725627013187874313445401912, −8.462696968055348274953213024643, −7.987425745198345241223595938623, −6.91565713363923914392457449652, −6.44334854620729196704432493871, −5.11129558682569747101636526948, −4.41410227210658894190443191545, −3.68874342306983489244066282305, −2.35318901422880603439814620623, −0.61354939182090444361538852552,
0.61354939182090444361538852552, 2.35318901422880603439814620623, 3.68874342306983489244066282305, 4.41410227210658894190443191545, 5.11129558682569747101636526948, 6.44334854620729196704432493871, 6.91565713363923914392457449652, 7.987425745198345241223595938623, 8.462696968055348274953213024643, 9.307725627013187874313445401912
