| L(s) = 1 | + 0.431·2-s − 1.81·4-s + 5-s − 2.42·7-s − 1.64·8-s + 0.431·10-s − 0.00755·11-s − 2.73·13-s − 1.04·14-s + 2.91·16-s + 4.73·17-s + 2.74·19-s − 1.81·20-s − 0.00325·22-s + 3.68·23-s + 25-s − 1.17·26-s + 4.39·28-s − 29-s + 1.25·31-s + 4.55·32-s + 2.04·34-s − 2.42·35-s + 6.60·37-s + 1.18·38-s − 1.64·40-s + 0.160·41-s + ⋯ |
| L(s) = 1 | + 0.305·2-s − 0.906·4-s + 0.447·5-s − 0.916·7-s − 0.581·8-s + 0.136·10-s − 0.00227·11-s − 0.757·13-s − 0.279·14-s + 0.729·16-s + 1.14·17-s + 0.629·19-s − 0.405·20-s − 0.000694·22-s + 0.769·23-s + 0.200·25-s − 0.231·26-s + 0.831·28-s − 0.185·29-s + 0.225·31-s + 0.804·32-s + 0.350·34-s − 0.409·35-s + 1.08·37-s + 0.192·38-s − 0.260·40-s + 0.0250·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.418353556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418353556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.431T + 2T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 + 0.00755T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 - 6.60T + 37T^{2} \) |
| 41 | \( 1 - 0.160T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 4.69T + 67T^{2} \) |
| 71 | \( 1 - 1.48T + 71T^{2} \) |
| 73 | \( 1 - 2.75T + 73T^{2} \) |
| 79 | \( 1 - 2.72T + 79T^{2} \) |
| 83 | \( 1 + 7.81T + 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 + 7.02T + 97T^{2} \) |
| 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
−9.579621732537984826438462580789, −9.150471074847902564009627814376, −8.013170310112582796775189720991, −7.20565407356210053359518570791, −6.09659681256263149858805649417, −5.46529441741522572718183172847, −4.58301396687217729650998958456, −3.50627935372303529769963027706, −2.69670009959084928639535592460, −0.847877574432523795244460143323,
0.847877574432523795244460143323, 2.69670009959084928639535592460, 3.50627935372303529769963027706, 4.58301396687217729650998958456, 5.46529441741522572718183172847, 6.09659681256263149858805649417, 7.20565407356210053359518570791, 8.013170310112582796775189720991, 9.150471074847902564009627814376, 9.579621732537984826438462580789
