| L(s) = 1 | − 1.80·2-s + 1.39·3-s + 1.24·4-s − 1.40·5-s − 2.51·6-s + 3.53·7-s + 1.35·8-s − 1.04·9-s + 2.52·10-s + 6.31·11-s + 1.74·12-s − 0.107·13-s − 6.36·14-s − 1.96·15-s − 4.93·16-s + 2.51·17-s + 1.88·18-s − 3.12·19-s − 1.74·20-s + 4.93·21-s − 11.3·22-s − 0.715·23-s + 1.89·24-s − 3.02·25-s + 0.193·26-s − 5.65·27-s + 4.40·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.806·3-s + 0.622·4-s − 0.627·5-s − 1.02·6-s + 1.33·7-s + 0.480·8-s − 0.349·9-s + 0.799·10-s + 1.90·11-s + 0.502·12-s − 0.0297·13-s − 1.70·14-s − 0.506·15-s − 1.23·16-s + 0.610·17-s + 0.444·18-s − 0.716·19-s − 0.391·20-s + 1.07·21-s − 2.42·22-s − 0.149·23-s + 0.387·24-s − 0.605·25-s + 0.0378·26-s − 1.08·27-s + 0.831·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131913276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131913276\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 3 | \( 1 - 1.39T + 3T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 11 | \( 1 - 6.31T + 11T^{2} \) |
| 13 | \( 1 + 0.107T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 0.715T + 23T^{2} \) |
| 31 | \( 1 - 5.88T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 1.54T + 43T^{2} \) |
| 47 | \( 1 - 8.45T + 47T^{2} \) |
| 53 | \( 1 - 7.16T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 0.706T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 0.291T + 83T^{2} \) |
| 89 | \( 1 - 8.90T + 89T^{2} \) |
| 97 | \( 1 + 3.62T + 97T^{2} \) |
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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
−9.919302454799454674842466820901, −9.068818930443005956629365004052, −8.583277769575542850006955293976, −7.932970711658753698421671444060, −7.32809438260378461988532477288, −6.08218700706782918950204605971, −4.52923663963947328185807591770, −3.81949900890107972198674256022, −2.19635691207899428415085200517, −1.09515077037431412824611664468,
1.09515077037431412824611664468, 2.19635691207899428415085200517, 3.81949900890107972198674256022, 4.52923663963947328185807591770, 6.08218700706782918950204605971, 7.32809438260378461988532477288, 7.932970711658753698421671444060, 8.583277769575542850006955293976, 9.068818930443005956629365004052, 9.919302454799454674842466820901
