L(s) = 1 | − 3-s − 5-s + 9-s − 11-s − 2·13-s + 15-s + 2·17-s + 4·19-s + 25-s − 27-s − 2·29-s + 33-s − 2·37-s + 2·39-s + 6·41-s − 45-s + 8·47-s − 2·51-s − 2·53-s + 55-s − 4·57-s + 4·59-s + 10·61-s + 2·65-s − 4·67-s − 14·73-s − 75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.174·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.149·45-s + 1.16·47-s − 0.280·51-s − 0.274·53-s + 0.134·55-s − 0.529·57-s + 0.520·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s − 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.424036480\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.424036480\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.45294990761779, −12.76251951996385, −12.63549584143231, −11.95229045786052, −11.55965272730151, −11.25541808447747, −10.57639111002032, −10.05376919427949, −9.847417008953057, −9.013413770808928, −8.723494508709439, −7.918962150425797, −7.441342024765026, −7.254228181899761, −6.558666872521901, −5.822377379403912, −5.529779842683628, −4.967141735707707, −4.356923630640543, −3.879427929304650, −3.146339943601476, −2.654192806946666, −1.845616692463873, −1.073408485330433, −0.4255992206132463,
0.4255992206132463, 1.073408485330433, 1.845616692463873, 2.654192806946666, 3.146339943601476, 3.879427929304650, 4.356923630640543, 4.967141735707707, 5.529779842683628, 5.822377379403912, 6.558666872521901, 7.254228181899761, 7.441342024765026, 7.918962150425797, 8.723494508709439, 9.013413770808928, 9.847417008953057, 10.05376919427949, 10.57639111002032, 11.25541808447747, 11.55965272730151, 11.95229045786052, 12.63549584143231, 12.76251951996385, 13.45294990761779