| L(s) = 1 | − 2-s + 0.784·3-s + 4-s − 5-s − 0.784·6-s + 1.82·7-s − 8-s − 2.38·9-s + 10-s + 4.31·11-s + 0.784·12-s + 5.09·13-s − 1.82·14-s − 0.784·15-s + 16-s + 3.84·17-s + 2.38·18-s − 5.69·19-s − 20-s + 1.43·21-s − 4.31·22-s + 8.76·23-s − 0.784·24-s + 25-s − 5.09·26-s − 4.22·27-s + 1.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.453·3-s + 0.5·4-s − 0.447·5-s − 0.320·6-s + 0.691·7-s − 0.353·8-s − 0.794·9-s + 0.316·10-s + 1.29·11-s + 0.226·12-s + 1.41·13-s − 0.488·14-s − 0.202·15-s + 0.250·16-s + 0.932·17-s + 0.561·18-s − 1.30·19-s − 0.223·20-s + 0.313·21-s − 0.919·22-s + 1.82·23-s − 0.160·24-s + 0.200·25-s − 0.999·26-s − 0.813·27-s + 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.638355073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638355073\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 223 | \( 1 - T \) |
| good | 3 | \( 1 - 0.784T + 3T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 - 3.84T + 17T^{2} \) |
| 19 | \( 1 + 5.69T + 19T^{2} \) |
| 23 | \( 1 - 8.76T + 23T^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 + 5.64T + 31T^{2} \) |
| 37 | \( 1 - 1.94T + 37T^{2} \) |
| 41 | \( 1 - 1.97T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 7.71T + 67T^{2} \) |
| 71 | \( 1 - 8.84T + 71T^{2} \) |
| 73 | \( 1 + 2.29T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 6.22T + 97T^{2} \) |
| 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
−8.983875687215023774382979228234, −8.337939058393808271450993561049, −7.81496209688094435798229584077, −6.79161864351433991816760884513, −6.12117200762085605647422507086, −5.11127288531756608158612633300, −3.85939109865586125106841024676, −3.30109953622184619990634970887, −1.95964887739689678342218725550, −0.960120751938482228088285269763,
0.960120751938482228088285269763, 1.95964887739689678342218725550, 3.30109953622184619990634970887, 3.85939109865586125106841024676, 5.11127288531756608158612633300, 6.12117200762085605647422507086, 6.79161864351433991816760884513, 7.81496209688094435798229584077, 8.337939058393808271450993561049, 8.983875687215023774382979228234
