L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 4.91·7-s + 8-s + 9-s − 10-s + 3.76·11-s + 12-s + 1.24·13-s + 4.91·14-s − 15-s + 16-s + 18-s + 5.93·19-s − 20-s + 4.91·21-s + 3.76·22-s − 0.0278·23-s + 24-s + 25-s + 1.24·26-s + 27-s + 4.91·28-s − 8.43·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.85·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.13·11-s + 0.288·12-s + 0.346·13-s + 1.31·14-s − 0.258·15-s + 0.250·16-s + 0.235·18-s + 1.36·19-s − 0.223·20-s + 1.07·21-s + 0.803·22-s − 0.00580·23-s + 0.204·24-s + 0.200·25-s + 0.244·26-s + 0.192·27-s + 0.928·28-s − 1.56·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.903947956\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.903947956\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 - 4.91T + 7T^{2} \) |
| 11 | \( 1 - 3.76T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 + 0.0278T + 23T^{2} \) |
| 29 | \( 1 + 8.43T + 29T^{2} \) |
| 31 | \( 1 - 9.60T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 + 8.30T + 47T^{2} \) |
| 53 | \( 1 + 1.77T + 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 8.38T + 67T^{2} \) |
| 71 | \( 1 - 3.51T + 71T^{2} \) |
| 73 | \( 1 + 8.51T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 + 2.10T + 97T^{2} \) |
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
−7.82623298482559079463666542321, −7.11670469726596271029235656096, −6.49578273629317064713143033482, −5.34868682310531537586455256355, −5.03633656195830334420180308735, −4.08363302144025000173796680177, −3.73333066139366578723071125908, −2.75353449016575919242295444198, −1.68779308785905742748565266301, −1.21438407863335915143944798553,
1.21438407863335915143944798553, 1.68779308785905742748565266301, 2.75353449016575919242295444198, 3.73333066139366578723071125908, 4.08363302144025000173796680177, 5.03633656195830334420180308735, 5.34868682310531537586455256355, 6.49578273629317064713143033482, 7.11670469726596271029235656096, 7.82623298482559079463666542321