L(s) = 1 | + 2-s + 3-s + 4-s − 3.23·5-s + 6-s + 4.29·7-s + 8-s + 9-s − 3.23·10-s + 11-s + 12-s + 2.88·13-s + 4.29·14-s − 3.23·15-s + 16-s − 1.40·17-s + 18-s + 3.85·19-s − 3.23·20-s + 4.29·21-s + 22-s + 2.00·23-s + 24-s + 5.45·25-s + 2.88·26-s + 27-s + 4.29·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.44·5-s + 0.408·6-s + 1.62·7-s + 0.353·8-s + 0.333·9-s − 1.02·10-s + 0.301·11-s + 0.288·12-s + 0.798·13-s + 1.14·14-s − 0.834·15-s + 0.250·16-s − 0.341·17-s + 0.235·18-s + 0.885·19-s − 0.722·20-s + 0.936·21-s + 0.213·22-s + 0.417·23-s + 0.204·24-s + 1.09·25-s + 0.564·26-s + 0.192·27-s + 0.811·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.894360487\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.894360487\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 4.29T + 7T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 - 0.769T + 31T^{2} \) |
| 37 | \( 1 - 6.59T + 37T^{2} \) |
| 41 | \( 1 + 7.18T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 0.896T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 67 | \( 1 - 6.74T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 - 0.751T + 73T^{2} \) |
| 79 | \( 1 - 0.132T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 2.50T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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
−8.270861129180040910594834652319, −7.76907053075966367277562140838, −7.17188373758616414097430871252, −6.27924542855360796254552904000, −5.03299391848282277997241427487, −4.70026144641613899486069008410, −3.76673030255661051800391767743, −3.32611109228853758071036621066, −2.04495510877173696622475967223, −1.07464930759986874719253602787,
1.07464930759986874719253602787, 2.04495510877173696622475967223, 3.32611109228853758071036621066, 3.76673030255661051800391767743, 4.70026144641613899486069008410, 5.03299391848282277997241427487, 6.27924542855360796254552904000, 7.17188373758616414097430871252, 7.76907053075966367277562140838, 8.270861129180040910594834652319