L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 4·11-s − 2·13-s + 14-s + 16-s − 2·17-s + 4·19-s − 20-s − 4·22-s + 8·23-s + 25-s + 2·26-s − 28-s + 2·29-s − 32-s + 2·34-s + 35-s + 6·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.986·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9951275620\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9951275620\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.57709798673782268275028132549, −9.437870818355534879168031071986, −9.118954839776025628038155273468, −7.983671143656830403181428322838, −7.07941601012087388140383379015, −6.44305888127139518988332786048, −5.09599254318882530046900067399, −3.85422726721423423764474467794, −2.69678884580615531725956839791, −0.996088059894723745963619169753,
0.996088059894723745963619169753, 2.69678884580615531725956839791, 3.85422726721423423764474467794, 5.09599254318882530046900067399, 6.44305888127139518988332786048, 7.07941601012087388140383379015, 7.983671143656830403181428322838, 9.118954839776025628038155273468, 9.437870818355534879168031071986, 10.57709798673782268275028132549