L(s) = 1 | − 2-s − 4-s + 3·8-s − 4·11-s − 2·13-s − 16-s + 6·17-s − 4·19-s + 4·22-s + 2·26-s + 2·29-s − 5·32-s − 6·34-s − 6·37-s + 4·38-s + 2·41-s + 4·43-s + 4·44-s + 2·52-s + 6·53-s − 2·58-s + 12·59-s + 2·61-s + 7·64-s − 4·67-s − 6·68-s − 6·73-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.20·11-s − 0.554·13-s − 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.852·22-s + 0.392·26-s + 0.371·29-s − 0.883·32-s − 1.02·34-s − 0.986·37-s + 0.648·38-s + 0.312·41-s + 0.609·43-s + 0.603·44-s + 0.277·52-s + 0.824·53-s − 0.262·58-s + 1.56·59-s + 0.256·61-s + 7/8·64-s − 0.488·67-s − 0.727·68-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.99069065724285, −16.22710142713776, −15.89073016524156, −15.03396422576230, −14.47390705378952, −14.02280140656004, −13.22550943438158, −12.85786996105836, −12.25052365292911, −11.54501906359632, −10.67472283981550, −10.15602740765412, −10.02940919027951, −9.038635019553541, −8.594781001669159, −7.874736694116864, −7.534453346505802, −6.791961467249262, −5.708198359986032, −5.273089213370730, −4.550782535257948, −3.791314278881566, −2.864496644648222, −2.029589325995463, −0.9575133710898381, 0,
0.9575133710898381, 2.029589325995463, 2.864496644648222, 3.791314278881566, 4.550782535257948, 5.273089213370730, 5.708198359986032, 6.791961467249262, 7.534453346505802, 7.874736694116864, 8.594781001669159, 9.038635019553541, 10.02940919027951, 10.15602740765412, 10.67472283981550, 11.54501906359632, 12.25052365292911, 12.85786996105836, 13.22550943438158, 14.02280140656004, 14.47390705378952, 15.03396422576230, 15.89073016524156, 16.22710142713776, 16.99069065724285