L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 4·17-s − 18-s + 5·19-s + 20-s + 21-s + 22-s − 3·23-s + 24-s + 25-s + 26-s − 27-s − 28-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 − 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.568434605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.568434605\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.56021843626971, −12.27821738860356, −11.70910615210332, −11.39227618810657, −10.78971139697334, −10.31280874006894, −9.904559385748274, −9.667404386444836, −9.235445139736999, −8.498510937835146, −8.018660918171991, −7.711235479853055, −7.089574402465536, −6.527498294439656, −6.320352244685579, −5.630290841838762, −5.099551998519352, −4.931020625276865, −3.936872348038176, −3.394792639816313, −2.914267998391374, −2.252838240058744, −1.633133324698668, −0.9747747726866876, −0.4631674831098776,
0.4631674831098776, 0.9747747726866876, 1.633133324698668, 2.252838240058744, 2.914267998391374, 3.394792639816313, 3.936872348038176, 4.931020625276865, 5.099551998519352, 5.630290841838762, 6.320352244685579, 6.527498294439656, 7.089574402465536, 7.711235479853055, 8.018660918171991, 8.498510937835146, 9.235445139736999, 9.667404386444836, 9.904559385748274, 10.31280874006894, 10.78971139697334, 11.39227618810657, 11.70910615210332, 12.27821738860356, 12.56021843626971