| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 6·11-s + 12-s + 3·13-s + 14-s + 15-s + 16-s + 18-s + 2·19-s + 20-s + 21-s − 6·22-s + 4·23-s + 24-s + 25-s + 3·26-s + 27-s + 28-s + 4·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.218·21-s − 1.27·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.192·27-s + 0.188·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.622851963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.622851963\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.81233007713549, −12.16417965554363, −11.58588876952163, −11.12783974848505, −10.70434555884119, −10.27596472504814, −9.912739873943369, −9.385028658464800, −8.673520240372035, −8.325778019822848, −7.957068600348483, −7.533464677577906, −6.837940214742532, −6.492313383446214, −5.974137325783803, −5.279373307999800, −4.901926220480718, −4.796531714003740, −3.849557990101609, −3.339582188926140, −2.936425540509597, −2.419275593773767, −1.934538043096709, −1.224183010079247, −0.6055808006591705,
0.6055808006591705, 1.224183010079247, 1.934538043096709, 2.419275593773767, 2.936425540509597, 3.339582188926140, 3.849557990101609, 4.796531714003740, 4.901926220480718, 5.279373307999800, 5.974137325783803, 6.492313383446214, 6.837940214742532, 7.533464677577906, 7.957068600348483, 8.325778019822848, 8.673520240372035, 9.385028658464800, 9.912739873943369, 10.27596472504814, 10.70434555884119, 11.12783974848505, 11.58588876952163, 12.16417965554363, 12.81233007713549
