L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 1.61·7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 3.61·13-s − 1.61·14-s + 15-s + 16-s + 17-s + 18-s + 3.43·19-s + 20-s − 1.61·21-s + 22-s − 5.27·23-s + 24-s + 25-s + 3.61·26-s + 27-s − 1.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.611·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 1.00·13-s − 0.432·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.787·19-s + 0.223·20-s − 0.353·21-s + 0.213·22-s − 1.09·23-s + 0.204·24-s + 0.200·25-s + 0.709·26-s + 0.192·27-s − 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.560094210\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.560094210\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 1.61T + 7T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 + 6.81T + 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 - 9.32T + 37T^{2} \) |
| 41 | \( 1 + 0.996T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 59 | \( 1 - 4.13T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 + 0.328T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 - 8.92T + 79T^{2} \) |
| 83 | \( 1 - 8.72T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 6.37T + 97T^{2} \) |
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
−7.979301659122491477115204420471, −7.45374176827522819780237976430, −6.49924454530163819917942643037, −6.02073636511943727101557457403, −5.36333887968497969106842322242, −4.23605849450824056652983535233, −3.71353187698148540983936161841, −2.93408212814569450022812408960, −2.08832732632262895135142922351, −1.05076889562059179320160805672,
1.05076889562059179320160805672, 2.08832732632262895135142922351, 2.93408212814569450022812408960, 3.71353187698148540983936161841, 4.23605849450824056652983535233, 5.36333887968497969106842322242, 6.02073636511943727101557457403, 6.49924454530163819917942643037, 7.45374176827522819780237976430, 7.979301659122491477115204420471