L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 7-s − 8-s + 9-s − 11-s + 2·12-s − 2·13-s + 14-s + 16-s − 6·17-s − 18-s + 2·19-s − 2·21-s + 22-s + 6·23-s − 2·24-s + 2·26-s − 4·27-s − 28-s + 8·31-s − 32-s − 2·33-s + 6·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.408·24-s + 0.392·26-s − 0.769·27-s − 0.188·28-s + 1.43·31-s − 0.176·32-s − 0.348·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.431423824681768623527587486128, −7.42095515728907715311240384454, −7.00306196646652234350789106232, −6.10665953853877978554321912136, −5.05359347783379215708500589274, −4.13383336531596327906907292177, −2.94782016582466902477791723277, −2.68589617967447541275595369222, −1.54438612546072859799086497537, 0,
1.54438612546072859799086497537, 2.68589617967447541275595369222, 2.94782016582466902477791723277, 4.13383336531596327906907292177, 5.05359347783379215708500589274, 6.10665953853877978554321912136, 7.00306196646652234350789106232, 7.42095515728907715311240384454, 8.431423824681768623527587486128