| L(s) = 1 | − 2-s − 0.0770·3-s + 4-s + 5-s + 0.0770·6-s − 1.82·7-s − 8-s − 2.99·9-s − 10-s − 3.11·11-s − 0.0770·12-s − 7.04·13-s + 1.82·14-s − 0.0770·15-s + 16-s − 2.97·17-s + 2.99·18-s + 5.65·19-s + 20-s + 0.140·21-s + 3.11·22-s − 5.92·23-s + 0.0770·24-s + 25-s + 7.04·26-s + 0.461·27-s − 1.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0444·3-s + 0.5·4-s + 0.447·5-s + 0.0314·6-s − 0.691·7-s − 0.353·8-s − 0.998·9-s − 0.316·10-s − 0.940·11-s − 0.0222·12-s − 1.95·13-s + 0.488·14-s − 0.0198·15-s + 0.250·16-s − 0.722·17-s + 0.705·18-s + 1.29·19-s + 0.223·20-s + 0.0307·21-s + 0.665·22-s − 1.23·23-s + 0.0157·24-s + 0.200·25-s + 1.38·26-s + 0.0888·27-s − 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4420575999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4420575999\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0770T + 3T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 + 7.04T + 13T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.92T + 23T^{2} \) |
| 29 | \( 1 - 7.96T + 29T^{2} \) |
| 31 | \( 1 - 2.90T + 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 + 0.616T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 2.95T + 47T^{2} \) |
| 53 | \( 1 - 4.08T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 4.05T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 5.06T + 73T^{2} \) |
| 79 | \( 1 + 1.95T + 79T^{2} \) |
| 83 | \( 1 + 3.26T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 2.28T + 97T^{2} \) |
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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
−8.102455063831936452279753930429, −7.47521992134872274559970881729, −6.70731618876458419981924915018, −6.08904694860306566848732605046, −5.23026912341492536452329568850, −4.72124041201180290150048711196, −3.17148979423790735214906348420, −2.74911815205620143814910416559, −1.94858220167066511793419466523, −0.35975393081454196179121306396,
0.35975393081454196179121306396, 1.94858220167066511793419466523, 2.74911815205620143814910416559, 3.17148979423790735214906348420, 4.72124041201180290150048711196, 5.23026912341492536452329568850, 6.08904694860306566848732605046, 6.70731618876458419981924915018, 7.47521992134872274559970881729, 8.102455063831936452279753930429
