L(s) = 1 | + 3.09·3-s + 1.68·5-s + 7-s + 6.56·9-s + 1.03·11-s − 4.59·13-s + 5.21·15-s − 1.50·17-s − 7.62·19-s + 3.09·21-s + 23-s − 2.15·25-s + 11.0·27-s − 1.99·29-s − 6.28·31-s + 3.19·33-s + 1.68·35-s + 7.43·37-s − 14.2·39-s + 2.41·41-s + 9.75·43-s + 11.0·45-s − 5.47·47-s + 49-s − 4.65·51-s − 14.0·53-s + 1.74·55-s + ⋯ |
L(s) = 1 | + 1.78·3-s + 0.754·5-s + 0.377·7-s + 2.18·9-s + 0.311·11-s − 1.27·13-s + 1.34·15-s − 0.365·17-s − 1.74·19-s + 0.674·21-s + 0.208·23-s − 0.430·25-s + 2.12·27-s − 0.370·29-s − 1.12·31-s + 0.555·33-s + 0.285·35-s + 1.22·37-s − 2.27·39-s + 0.376·41-s + 1.48·43-s + 1.65·45-s − 0.798·47-s + 0.142·49-s − 0.652·51-s − 1.92·53-s + 0.234·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.939229813\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.939229813\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + 7.62T + 19T^{2} \) |
| 29 | \( 1 + 1.99T + 29T^{2} \) |
| 31 | \( 1 + 6.28T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 2.41T + 41T^{2} \) |
| 43 | \( 1 - 9.75T + 43T^{2} \) |
| 47 | \( 1 + 5.47T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 0.248T + 61T^{2} \) |
| 67 | \( 1 - 6.60T + 67T^{2} \) |
| 71 | \( 1 + 3.58T + 71T^{2} \) |
| 73 | \( 1 - 2.21T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−10.23054556414928730682674972284, −9.463451754443065556075853560327, −8.965601261938094561598799435561, −8.014447996932126343988249518933, −7.31425172924259722510601788242, −6.22704745559759103167544381516, −4.76790115555116901445658646274, −3.86176491088755739329487811105, −2.49072573921677739459001514456, −1.93304379953186733955596165812,
1.93304379953186733955596165812, 2.49072573921677739459001514456, 3.86176491088755739329487811105, 4.76790115555116901445658646274, 6.22704745559759103167544381516, 7.31425172924259722510601788242, 8.014447996932126343988249518933, 8.965601261938094561598799435561, 9.463451754443065556075853560327, 10.23054556414928730682674972284