| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 2.56·7-s + 8-s + 9-s + 10-s + 2.56·11-s − 12-s + 2·13-s + 2.56·14-s − 15-s + 16-s − 3.12·17-s + 18-s − 7.68·19-s + 20-s − 2.56·21-s + 2.56·22-s + 1.43·23-s − 24-s + 25-s + 2·26-s − 27-s + 2.56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.968·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.772·11-s − 0.288·12-s + 0.554·13-s + 0.684·14-s − 0.258·15-s + 0.250·16-s − 0.757·17-s + 0.235·18-s − 1.76·19-s + 0.223·20-s − 0.558·21-s + 0.546·22-s + 0.299·23-s − 0.204·24-s + 0.200·25-s + 0.392·26-s − 0.192·27-s + 0.484·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.531019911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.531019911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 - 7.43T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 4.31T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.58369742668169622881392498137, −9.135189465923416730781059868944, −8.454113429484776346681778764311, −7.27109803947625248603649878617, −6.36442623568392223854921019561, −5.82657404658869598588534061457, −4.59322043530120736241495437638, −4.18573751729956530611062536605, −2.50779933605166344714653458320, −1.35636171358021831321444547883,
1.35636171358021831321444547883, 2.50779933605166344714653458320, 4.18573751729956530611062536605, 4.59322043530120736241495437638, 5.82657404658869598588534061457, 6.36442623568392223854921019561, 7.27109803947625248603649878617, 8.454113429484776346681778764311, 9.135189465923416730781059868944, 10.58369742668169622881392498137
