| L(s) = 1 | + 2-s − 0.755·3-s + 4-s − 3.67·5-s − 0.755·6-s − 3.29·7-s + 8-s − 2.42·9-s − 3.67·10-s + 1.31·11-s − 0.755·12-s + 5.07·13-s − 3.29·14-s + 2.77·15-s + 16-s − 1.19·17-s − 2.42·18-s − 0.339·19-s − 3.67·20-s + 2.49·21-s + 1.31·22-s + 3.05·23-s − 0.755·24-s + 8.51·25-s + 5.07·26-s + 4.10·27-s − 3.29·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.436·3-s + 0.5·4-s − 1.64·5-s − 0.308·6-s − 1.24·7-s + 0.353·8-s − 0.809·9-s − 1.16·10-s + 0.395·11-s − 0.218·12-s + 1.40·13-s − 0.880·14-s + 0.717·15-s + 0.250·16-s − 0.290·17-s − 0.572·18-s − 0.0778·19-s − 0.822·20-s + 0.543·21-s + 0.279·22-s + 0.636·23-s − 0.154·24-s + 1.70·25-s + 0.995·26-s + 0.789·27-s − 0.622·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.231227913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.231227913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 0.755T + 3T^{2} \) |
| 5 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 + 0.339T + 19T^{2} \) |
| 23 | \( 1 - 3.05T + 23T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 - 3.86T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 - 2.30T + 43T^{2} \) |
| 47 | \( 1 + 1.69T + 47T^{2} \) |
| 53 | \( 1 - 0.200T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 3.92T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 7.54T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.379357081371792186171689512045, −8.576156414879909017734872160296, −7.82527856012800750876340551889, −6.73544020958614848378399399795, −6.34288141746426062860353896199, −5.35037253505882936980050957684, −4.18720904434357161073177364453, −3.60172010726495328162005901926, −2.85447298834612286773413453482, −0.70850588558138142425695982539,
0.70850588558138142425695982539, 2.85447298834612286773413453482, 3.60172010726495328162005901926, 4.18720904434357161073177364453, 5.35037253505882936980050957684, 6.34288141746426062860353896199, 6.73544020958614848378399399795, 7.82527856012800750876340551889, 8.576156414879909017734872160296, 9.379357081371792186171689512045
