| L(s) = 1 | − 3·3-s + 3·7-s + 6·9-s + 11-s − 5·13-s + 4·17-s − 8·19-s − 9·21-s + 7·23-s − 5·25-s − 9·27-s + 6·29-s − 2·31-s − 3·33-s + 10·37-s + 15·39-s − 6·41-s + 43-s − 7·47-s + 2·49-s − 12·51-s − 6·53-s + 24·57-s − 4·59-s + 7·61-s + 18·63-s − 7·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.13·7-s + 2·9-s + 0.301·11-s − 1.38·13-s + 0.970·17-s − 1.83·19-s − 1.96·21-s + 1.45·23-s − 25-s − 1.73·27-s + 1.11·29-s − 0.359·31-s − 0.522·33-s + 1.64·37-s + 2.40·39-s − 0.937·41-s + 0.152·43-s − 1.02·47-s + 2/7·49-s − 1.68·51-s − 0.824·53-s + 3.17·57-s − 0.520·59-s + 0.896·61-s + 2.26·63-s − 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9905011958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9905011958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.13139863011179, −14.55097955553478, −14.21321638164051, −13.21903415915683, −12.71471538875933, −12.33313191718377, −11.81034743275675, −11.23914367518038, −11.06075294437516, −10.30536035618734, −9.914252403259897, −9.314403094767352, −8.361698900632814, −7.926624655795490, −7.251265612030200, −6.674203174796419, −6.208785007532943, −5.471378208232533, −4.972697305754041, −4.602026888654877, −4.042163364468582, −2.890904277182950, −1.975969328287918, −1.305515441423727, −0.4511079818514394,
0.4511079818514394, 1.305515441423727, 1.975969328287918, 2.890904277182950, 4.042163364468582, 4.602026888654877, 4.972697305754041, 5.471378208232533, 6.208785007532943, 6.674203174796419, 7.251265612030200, 7.926624655795490, 8.361698900632814, 9.314403094767352, 9.914252403259897, 10.30536035618734, 11.06075294437516, 11.23914367518038, 11.81034743275675, 12.33313191718377, 12.71471538875933, 13.21903415915683, 14.21321638164051, 14.55097955553478, 15.13139863011179
