L(s) = 1 | + 2·7-s + 4·11-s − 13-s − 6·17-s − 6·19-s + 2·29-s + 6·31-s − 10·37-s − 8·41-s + 12·43-s − 12·47-s − 3·49-s − 6·53-s + 2·61-s + 2·67-s − 8·71-s − 14·73-s + 8·77-s − 4·79-s − 8·83-s − 4·89-s − 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 1.20·11-s − 0.277·13-s − 1.45·17-s − 1.37·19-s + 0.371·29-s + 1.07·31-s − 1.64·37-s − 1.24·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.256·61-s + 0.244·67-s − 0.949·71-s − 1.63·73-s + 0.911·77-s − 0.450·79-s − 0.878·83-s − 0.423·89-s − 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.638202837\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.638202837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.97709599710834, −13.28702346031872, −12.90783966853489, −12.25972991825016, −11.85790878880403, −11.31248655578064, −11.02420406706518, −10.34040872049776, −9.966385157712933, −9.196185004011412, −8.756963237717244, −8.449521624538284, −7.913429351844918, −7.072740641705274, −6.712733504306135, −6.325724711832618, −5.659552617585815, −4.825770940495532, −4.484704803693864, −4.102624895539043, −3.271458419425881, −2.598290396469973, −1.760182302849491, −1.556490031359764, −0.3896477726043913,
0.3896477726043913, 1.556490031359764, 1.760182302849491, 2.598290396469973, 3.271458419425881, 4.102624895539043, 4.484704803693864, 4.825770940495532, 5.659552617585815, 6.325724711832618, 6.712733504306135, 7.072740641705274, 7.913429351844918, 8.449521624538284, 8.756963237717244, 9.196185004011412, 9.966385157712933, 10.34040872049776, 11.02420406706518, 11.31248655578064, 11.85790878880403, 12.25972991825016, 12.90783966853489, 13.28702346031872, 13.97709599710834