L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s + 5·13-s + 14-s + 16-s − 18-s − 4·19-s + 21-s − 22-s − 6·23-s + 24-s − 5·26-s − 27-s − 28-s − 4·29-s + 3·31-s − 32-s − 33-s + 36-s + 4·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.188·28-s − 0.742·29-s + 0.538·31-s − 0.176·32-s − 0.174·33-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.246793059\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246793059\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.74586775183861, −12.02714093819613, −11.59817649381444, −11.37823315253230, −10.80610284302741, −10.37196455511301, −9.916244611364796, −9.628036733718482, −8.920061367823458, −8.483202293387910, −8.214997158748502, −7.633488244763199, −6.937624132221360, −6.551489367482305, −6.282934262821955, −5.636449973351384, −5.376361177581416, −4.363285910279242, −4.082744744560149, −3.514426082756071, −2.936893737090987, −2.022266833937198, −1.789855552375642, −0.9282810419371737, −0.4165121670349105,
0.4165121670349105, 0.9282810419371737, 1.789855552375642, 2.022266833937198, 2.936893737090987, 3.514426082756071, 4.082744744560149, 4.363285910279242, 5.376361177581416, 5.636449973351384, 6.282934262821955, 6.551489367482305, 6.937624132221360, 7.633488244763199, 8.214997158748502, 8.483202293387910, 8.920061367823458, 9.628036733718482, 9.916244611364796, 10.37196455511301, 10.80610284302741, 11.37823315253230, 11.59817649381444, 12.02714093819613, 12.74586775183861