| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s − 2·9-s + 10-s − 4·11-s + 12-s + 2·13-s − 2·14-s − 15-s + 16-s + 17-s + 2·18-s + 3·19-s − 20-s + 2·21-s + 4·22-s + 2·23-s − 24-s + 25-s − 2·26-s − 5·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 0.688·19-s − 0.223·20-s + 0.436·21-s + 0.852·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.962·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.97660674797347, −13.56496904095978, −13.11010100892021, −12.47477232115146, −11.86889089203203, −11.47559529764194, −10.94798862486512, −10.70244326161199, −9.991169562903723, −9.504139053549463, −8.779525560781037, −8.635383064751188, −7.937482280003678, −7.735128752915908, −7.254041556281592, −6.536344727294372, −5.807834944460942, −5.258370379328764, −4.956455169040052, −3.926721324897045, −3.440864765958081, −2.886796617504985, −2.259720381246186, −1.656839322839550, −0.8136349471412476, 0,
0.8136349471412476, 1.656839322839550, 2.259720381246186, 2.886796617504985, 3.440864765958081, 3.926721324897045, 4.956455169040052, 5.258370379328764, 5.807834944460942, 6.536344727294372, 7.254041556281592, 7.735128752915908, 7.937482280003678, 8.635383064751188, 8.779525560781037, 9.504139053549463, 9.991169562903723, 10.70244326161199, 10.94798862486512, 11.47559529764194, 11.86889089203203, 12.47477232115146, 13.11010100892021, 13.56496904095978, 13.97660674797347
