L(s) = 1 | − 2-s + 4-s − 3.23·5-s − 8-s + 3.23·10-s + 11-s + 16-s − 0.763·17-s − 5.70·19-s − 3.23·20-s − 22-s − 6.47·23-s + 5.47·25-s + 4.47·29-s + 7.23·31-s − 32-s + 0.763·34-s + 6.94·37-s + 5.70·38-s + 3.23·40-s − 0.763·41-s − 2.47·43-s + 44-s + 6.47·46-s − 9.70·47-s − 5.47·50-s + 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.44·5-s − 0.353·8-s + 1.02·10-s + 0.301·11-s + 0.250·16-s − 0.185·17-s − 1.30·19-s − 0.723·20-s − 0.213·22-s − 1.34·23-s + 1.09·25-s + 0.830·29-s + 1.29·31-s − 0.176·32-s + 0.131·34-s + 1.14·37-s + 0.925·38-s + 0.511·40-s − 0.119·41-s − 0.376·43-s + 0.150·44-s + 0.954·46-s − 1.41·47-s − 0.773·50-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5473479453\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5473479453\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 3.23T + 83T^{2} \) |
| 89 | \( 1 - 2.47T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.84765487368768860769242742653, −7.15020599873679668348541086871, −6.43818126420135066311204821204, −5.93481120913551524054522139116, −4.51043541353326318563307519364, −4.35898993130214179007988660286, −3.37851601074741963049161761425, −2.59968979450742003780057019075, −1.54866508644472456882636244626, −0.39589012665285907675282373958,
0.39589012665285907675282373958, 1.54866508644472456882636244626, 2.59968979450742003780057019075, 3.37851601074741963049161761425, 4.35898993130214179007988660286, 4.51043541353326318563307519364, 5.93481120913551524054522139116, 6.43818126420135066311204821204, 7.15020599873679668348541086871, 7.84765487368768860769242742653