L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s − 13-s + 16-s + 8·17-s − 18-s + 6·19-s − 4·22-s − 6·23-s + 24-s + 26-s − 27-s − 4·29-s − 32-s − 4·33-s − 8·34-s + 36-s + 2·37-s − 6·38-s + 39-s + 2·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 1.94·17-s − 0.235·18-s + 1.37·19-s − 0.852·22-s − 1.25·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.742·29-s − 0.176·32-s − 0.696·33-s − 1.37·34-s + 1/6·36-s + 0.328·37-s − 0.973·38-s + 0.160·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.785388416\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.785388416\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.84794995635352, −13.33609257513024, −12.57972610085808, −12.03811578497620, −11.80353524638011, −11.59959800722575, −10.67745184781601, −10.34181303107336, −9.797309320659902, −9.379185072505910, −9.057193474583628, −8.171832963629551, −7.689786858108362, −7.401793990800578, −6.769556199789096, −6.137942597437229, −5.637224630071349, −5.346728010207325, −4.408040460557867, −3.800698299305128, −3.335431208620378, −2.548759464057457, −1.669334634032827, −1.166073717451803, −0.5635371654678414,
0.5635371654678414, 1.166073717451803, 1.669334634032827, 2.548759464057457, 3.335431208620378, 3.800698299305128, 4.408040460557867, 5.346728010207325, 5.637224630071349, 6.137942597437229, 6.769556199789096, 7.401793990800578, 7.689786858108362, 8.171832963629551, 9.057193474583628, 9.379185072505910, 9.797309320659902, 10.34181303107336, 10.67745184781601, 11.59959800722575, 11.80353524638011, 12.03811578497620, 12.57972610085808, 13.33609257513024, 13.84794995635352