| L(s) = 1 | − 5-s − 3·7-s + 4·11-s − 3·17-s − 4·19-s − 23-s + 25-s − 29-s + 31-s + 3·35-s + 37-s − 3·41-s − 12·43-s + 10·47-s + 2·49-s + 9·53-s − 4·55-s + 9·59-s − 10·61-s + 67-s − 5·71-s − 12·77-s + 4·79-s + 9·83-s + 3·85-s + 8·89-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.20·11-s − 0.727·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.185·29-s + 0.179·31-s + 0.507·35-s + 0.164·37-s − 0.468·41-s − 1.82·43-s + 1.45·47-s + 2/7·49-s + 1.23·53-s − 0.539·55-s + 1.17·59-s − 1.28·61-s + 0.122·67-s − 0.593·71-s − 1.36·77-s + 0.450·79-s + 0.987·83-s + 0.325·85-s + 0.847·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174184319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174184319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−7.76364132546118976223330171554, −6.89169151718913384751733399228, −6.56219107141053445824983984564, −5.94697678821822871064527639127, −4.92244943626343921597989137252, −4.06056711558805015424167148621, −3.66295621882111115467321009261, −2.73915226491007000399655434742, −1.77746755409833811183434449578, −0.52204454276184848581613183790,
0.52204454276184848581613183790, 1.77746755409833811183434449578, 2.73915226491007000399655434742, 3.66295621882111115467321009261, 4.06056711558805015424167148621, 4.92244943626343921597989137252, 5.94697678821822871064527639127, 6.56219107141053445824983984564, 6.89169151718913384751733399228, 7.76364132546118976223330171554
