| L(s) = 1 | + 3-s + 7-s − 2·9-s − 11-s − 5·13-s + 6·17-s + 6·19-s + 21-s − 3·23-s − 5·25-s − 5·27-s − 29-s − 3·31-s − 33-s − 2·37-s − 5·39-s + 43-s + 49-s + 6·51-s − 5·53-s + 6·57-s + 4·59-s + 10·61-s − 2·63-s + 4·67-s − 3·69-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 1.38·13-s + 1.45·17-s + 1.37·19-s + 0.218·21-s − 0.625·23-s − 25-s − 0.962·27-s − 0.185·29-s − 0.538·31-s − 0.174·33-s − 0.328·37-s − 0.800·39-s + 0.152·43-s + 1/7·49-s + 0.840·51-s − 0.686·53-s + 0.794·57-s + 0.520·59-s + 1.28·61-s − 0.251·63-s + 0.488·67-s − 0.361·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.133127012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.133127012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 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
−13.68000889539836, −12.83114747855811, −12.40939838096626, −11.87922066825972, −11.66732275994622, −11.04741010852797, −10.42434820069149, −9.819359600499021, −9.596017780005234, −9.188283397421619, −8.369881548174800, −7.859759486936249, −7.775382474942237, −7.208897367423050, −6.529833786093626, −5.732421999560531, −5.302870370520434, −5.149817249270483, −4.221030549022011, −3.501238105813289, −3.291121029614585, −2.379500770914380, −2.163832690244409, −1.250239569283211, −0.4198092420936862,
0.4198092420936862, 1.250239569283211, 2.163832690244409, 2.379500770914380, 3.291121029614585, 3.501238105813289, 4.221030549022011, 5.149817249270483, 5.302870370520434, 5.732421999560531, 6.529833786093626, 7.208897367423050, 7.775382474942237, 7.859759486936249, 8.369881548174800, 9.188283397421619, 9.596017780005234, 9.819359600499021, 10.42434820069149, 11.04741010852797, 11.66732275994622, 11.87922066825972, 12.40939838096626, 12.83114747855811, 13.68000889539836
