| L(s) = 1 | − 2-s + 3-s + 4-s + 2.70·5-s − 6-s − 7-s − 8-s + 9-s − 2.70·10-s + 12-s − 0.701·13-s + 14-s + 2.70·15-s + 16-s − 18-s + 7.40·19-s + 2.70·20-s − 21-s − 23-s − 24-s + 2.29·25-s + 0.701·26-s + 27-s − 28-s + 6.70·29-s − 2.70·30-s + 6·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.20·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.854·10-s + 0.288·12-s − 0.194·13-s + 0.267·14-s + 0.697·15-s + 0.250·16-s − 0.235·18-s + 1.69·19-s + 0.604·20-s − 0.218·21-s − 0.208·23-s − 0.204·24-s + 0.459·25-s + 0.137·26-s + 0.192·27-s − 0.188·28-s + 1.24·29-s − 0.493·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.748393073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.748393073\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2.70T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.701T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.40T + 19T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 1.29T + 41T^{2} \) |
| 43 | \( 1 + 0.701T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2.59T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−9.860581681420956797531759186485, −9.365177163657749865299330200826, −8.493622714794371981303560464464, −7.61262486954545317966996722401, −6.70154313407607083989485715057, −5.90876413493259073290651989128, −4.86630425022447769465014053937, −3.29607028429502174270589481137, −2.43590585805091429662015644520, −1.23771561526534958407070322239,
1.23771561526534958407070322239, 2.43590585805091429662015644520, 3.29607028429502174270589481137, 4.86630425022447769465014053937, 5.90876413493259073290651989128, 6.70154313407607083989485715057, 7.61262486954545317966996722401, 8.493622714794371981303560464464, 9.365177163657749865299330200826, 9.860581681420956797531759186485
