| L(s) = 1 | − 3.46i·5-s + 3i·11-s − 3.46i·13-s − 3·17-s + 7i·19-s + 3.46·23-s − 6.99·25-s + 6.92i·29-s − 6.92·31-s − 10.3i·37-s + 3·41-s + 5i·43-s + 3.46·47-s − 7·49-s + 13.8i·53-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s + 0.904i·11-s − 0.960i·13-s − 0.727·17-s + 1.60i·19-s + 0.722·23-s − 1.39·25-s + 1.28i·29-s − 1.24·31-s − 1.70i·37-s + 0.468·41-s + 0.762i·43-s + 0.505·47-s − 49-s + 1.90i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9326895918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9326895918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 7iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 13.8iT - 53T^{2} \) |
| 59 | \( 1 - 9iT - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−8.540504210120491449259827136215, −7.55848869940702560738062767028, −7.23677342893132151854168217041, −5.79957031842222103767954373558, −5.60058498905582045767788420311, −4.59999369069931916001352670789, −4.15765653622779849855111491801, −3.06487457268308720718242879415, −1.84251033386763618022450503116, −1.12052016780289531714842326287,
0.25600603440507426508611179133, 1.91255533101006541385506483383, 2.74448070463133456375689840708, 3.36803981317194931125549267014, 4.27292813772160939265956156974, 5.15179618363935727224320485254, 6.16013017212697897248601824269, 6.75984698377778914902937001398, 7.04083620045951605320898076090, 8.020509725223781907428920466288
