L(s) = 1 | + (1 + 1.73i)2-s + (2.00 − 3.46i)4-s + (−2.5 + 4.33i)5-s + 24·8-s − 10·10-s + (−5 − 8.66i)11-s + (40 − 69.2i)13-s + (8.00 + 13.8i)16-s + 7·17-s − 113·19-s + (10 + 17.3i)20-s + (10 − 17.3i)22-s + (40.5 − 70.1i)23-s + (−12.5 − 21.6i)25-s + 160·26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 1.06·8-s − 0.316·10-s + (−0.137 − 0.237i)11-s + (0.853 − 1.47i)13-s + (0.125 + 0.216i)16-s + 0.0998·17-s − 1.36·19-s + (0.111 + 0.193i)20-s + (0.0969 − 0.167i)22-s + (0.367 − 0.635i)23-s + (−0.100 − 0.173i)25-s + 1.20·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.475642903\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.475642903\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (5 + 8.66i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-40 + 69.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 113T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-40.5 + 70.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-110 - 190. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-94.5 + 163. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 170T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-65 + 112. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (5 + 8.66i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (80 + 138. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 631T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-280 + 484. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (114.5 + 198. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (375 - 649. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 890T + 3.57e5T^{2} \) |
| 73 | \( 1 + 890T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-13.5 - 23.3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (214.5 + 371. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 750T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-740 - 1.28e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−10.57571434005830465618844711512, −10.29793437126315969448566650480, −8.673071436838610941183490654419, −7.908657646158297079983255914482, −6.82107133150000345965197748640, −6.06817113691749160782675534461, −5.16479484681170340555474660403, −3.91462561980045135461682604955, −2.52283878049274556591214733347, −0.797773692822905705911686711929,
1.39978303334069136820862239558, 2.60246916437341202100463681794, 4.00653498524167697162630630571, 4.55036436431716144292516809398, 6.17178170140818428417941749339, 7.13792582509322593793947896495, 8.210662595535854973225136133603, 8.991720875672659202206114653567, 10.19914843527716186449526683077, 11.14573202874587909315953686418