| L(s) = 1 | − 0.703·3-s − 0.200·5-s − 2.50·9-s + 0.369·11-s − 3.09·13-s + 0.141·15-s − 17-s + 3.18·19-s − 3.84·23-s − 4.95·25-s + 3.87·27-s − 2.87·29-s + 6.70·31-s − 0.259·33-s − 4.84·37-s + 2.17·39-s + 10.7·41-s + 12.5·43-s + 0.502·45-s − 0.142·47-s + 0.703·51-s + 7.07·53-s − 0.0740·55-s − 2.23·57-s − 6.71·59-s − 5.43·61-s + 0.620·65-s + ⋯ |
| L(s) = 1 | − 0.406·3-s − 0.0896·5-s − 0.835·9-s + 0.111·11-s − 0.858·13-s + 0.0364·15-s − 0.242·17-s + 0.730·19-s − 0.802·23-s − 0.991·25-s + 0.745·27-s − 0.533·29-s + 1.20·31-s − 0.0452·33-s − 0.796·37-s + 0.348·39-s + 1.68·41-s + 1.91·43-s + 0.0748·45-s − 0.0207·47-s + 0.0985·51-s + 0.971·53-s − 0.00998·55-s − 0.296·57-s − 0.874·59-s − 0.695·61-s + 0.0769·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.108509725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108509725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.703T + 3T^{2} \) |
| 5 | \( 1 + 0.200T + 5T^{2} \) |
| 11 | \( 1 - 0.369T + 11T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 19 | \( 1 - 3.18T + 19T^{2} \) |
| 23 | \( 1 + 3.84T + 23T^{2} \) |
| 29 | \( 1 + 2.87T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 + 4.84T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 0.142T + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 + 6.71T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 - 1.03T + 67T^{2} \) |
| 71 | \( 1 + 2.83T + 71T^{2} \) |
| 73 | \( 1 - 9.84T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 5.51T + 83T^{2} \) |
| 89 | \( 1 - 2.57T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.654732824494826196271615136424, −7.74706079664815259884451543263, −7.28834453139550542794320287628, −6.13123663781154831089336001237, −5.76561691624716390849781732567, −4.82691922256461606151447953315, −4.03704114848301523145750600403, −2.96364176519670467990755345573, −2.12185972904597207072534096091, −0.61832347012825404223812651152,
0.61832347012825404223812651152, 2.12185972904597207072534096091, 2.96364176519670467990755345573, 4.03704114848301523145750600403, 4.82691922256461606151447953315, 5.76561691624716390849781732567, 6.13123663781154831089336001237, 7.28834453139550542794320287628, 7.74706079664815259884451543263, 8.654732824494826196271615136424
