L(s) = 1 | + 2-s + 4-s − 3.47·5-s + 2.01·7-s + 8-s − 3.47·10-s − 5.41·11-s + 4.76·13-s + 2.01·14-s + 16-s + 0.975·17-s + 2.00·19-s − 3.47·20-s − 5.41·22-s + 4.93·23-s + 7.10·25-s + 4.76·26-s + 2.01·28-s + 4.34·29-s − 0.676·31-s + 32-s + 0.975·34-s − 7.01·35-s − 8.00·37-s + 2.00·38-s − 3.47·40-s − 5.01·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.55·5-s + 0.761·7-s + 0.353·8-s − 1.10·10-s − 1.63·11-s + 1.32·13-s + 0.538·14-s + 0.250·16-s + 0.236·17-s + 0.461·19-s − 0.778·20-s − 1.15·22-s + 1.02·23-s + 1.42·25-s + 0.933·26-s + 0.380·28-s + 0.806·29-s − 0.121·31-s + 0.176·32-s + 0.167·34-s − 1.18·35-s − 1.31·37-s + 0.326·38-s − 0.550·40-s − 0.783·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.397433284\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.397433284\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 + 3.47T + 5T^{2} \) |
| 7 | \( 1 - 2.01T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 - 0.975T + 17T^{2} \) |
| 19 | \( 1 - 2.00T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 - 4.34T + 29T^{2} \) |
| 31 | \( 1 + 0.676T + 31T^{2} \) |
| 37 | \( 1 + 8.00T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 + 2.02T + 43T^{2} \) |
| 47 | \( 1 + 9.42T + 47T^{2} \) |
| 53 | \( 1 + 2.40T + 53T^{2} \) |
| 59 | \( 1 - 6.80T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 0.737T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 0.894T + 89T^{2} \) |
| 97 | \( 1 + 6.58T + 97T^{2} \) |
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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
−7.87453363119047937675220175378, −7.18657077733581569911994357287, −6.53775585275100322331482503345, −5.40861368570834156616441653253, −5.02480374628316516962161421686, −4.35318402653063199216478752490, −3.37428179289641143462517623099, −3.14449395187175332380161893230, −1.84208644952441579919959073309, −0.69015862631680131573830427838,
0.69015862631680131573830427838, 1.84208644952441579919959073309, 3.14449395187175332380161893230, 3.37428179289641143462517623099, 4.35318402653063199216478752490, 5.02480374628316516962161421686, 5.40861368570834156616441653253, 6.53775585275100322331482503345, 7.18657077733581569911994357287, 7.87453363119047937675220175378