| L(s) = 1 | − 2·3-s − 14·5-s − 23·9-s − 20·11-s + 6·13-s + 28·15-s − 20·17-s − 102·19-s − 124·23-s + 71·25-s + 100·27-s − 78·29-s − 236·31-s + 40·33-s + 66·37-s − 12·39-s − 268·41-s + 132·43-s + 322·45-s − 516·47-s + 40·51-s − 354·53-s + 280·55-s + 204·57-s − 438·59-s − 486·61-s − 84·65-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 1.25·5-s − 0.851·9-s − 0.548·11-s + 0.128·13-s + 0.481·15-s − 0.285·17-s − 1.23·19-s − 1.12·23-s + 0.567·25-s + 0.712·27-s − 0.499·29-s − 1.36·31-s + 0.211·33-s + 0.293·37-s − 0.0492·39-s − 1.02·41-s + 0.468·43-s + 1.06·45-s − 1.60·47-s + 0.109·51-s − 0.917·53-s + 0.686·55-s + 0.474·57-s − 0.966·59-s − 1.02·61-s − 0.160·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.08610685180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08610685180\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 20 T + p^{3} T^{2} \) |
| 19 | \( 1 + 102 T + p^{3} T^{2} \) |
| 23 | \( 1 + 124 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 66 T + p^{3} T^{2} \) |
| 41 | \( 1 + 268 T + p^{3} T^{2} \) |
| 43 | \( 1 - 132 T + p^{3} T^{2} \) |
| 47 | \( 1 + 516 T + p^{3} T^{2} \) |
| 53 | \( 1 + 354 T + p^{3} T^{2} \) |
| 59 | \( 1 + 438 T + p^{3} T^{2} \) |
| 61 | \( 1 + 486 T + p^{3} T^{2} \) |
| 67 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 248 T + p^{3} T^{2} \) |
| 73 | \( 1 + 768 T + p^{3} T^{2} \) |
| 79 | \( 1 - 192 T + p^{3} T^{2} \) |
| 83 | \( 1 + 294 T + p^{3} T^{2} \) |
| 89 | \( 1 + 80 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1404 T + p^{3} T^{2} \) |
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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
−8.856291588164117553747481565634, −8.194003594577477419239962171608, −7.64081079990121259187377761596, −6.60464969952410463535165617791, −5.82973583266223099140261491300, −4.86636540788766152994946515786, −4.00398323453065357051399011333, −3.15601899876703821044943858598, −1.92925003525518874792606964564, −0.13656398868243097316839296795,
0.13656398868243097316839296795, 1.92925003525518874792606964564, 3.15601899876703821044943858598, 4.00398323453065357051399011333, 4.86636540788766152994946515786, 5.82973583266223099140261491300, 6.60464969952410463535165617791, 7.64081079990121259187377761596, 8.194003594577477419239962171608, 8.856291588164117553747481565634
