| L(s) = 1 | + (2.5 − 4.33i)5-s + (−18 − 31.1i)7-s + (−32 − 55.4i)11-s + (32.5 − 56.2i)13-s + 59·17-s − 28·19-s + (−80 + 138. i)23-s + (50 + 86.6i)25-s + (28.5 + 49.3i)29-s + (−82 + 142. i)31-s − 180.·35-s − 321·37-s + (123 − 213. i)41-s + (4 + 6.92i)43-s + (−42 − 72.7i)47-s + ⋯ |
| L(s) = 1 | + (0.223 − 0.387i)5-s + (−0.971 − 1.68i)7-s + (−0.877 − 1.51i)11-s + (0.693 − 1.20i)13-s + 0.841·17-s − 0.338·19-s + (−0.725 + 1.25i)23-s + (0.400 + 0.692i)25-s + (0.182 + 0.316i)29-s + (−0.475 + 0.822i)31-s − 0.869·35-s − 1.42·37-s + (0.468 − 0.811i)41-s + (0.0141 + 0.0245i)43-s + (−0.130 − 0.225i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8539788291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8539788291\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.5 + 4.33i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (18 + 31.1i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (32 + 55.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-32.5 + 56.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 59T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28T + 6.85e3T^{2} \) |
| 23 | \( 1 + (80 - 138. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-28.5 - 49.3i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (82 - 142. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 321T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-123 + 213. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (42 + 72.7i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 478T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-16 + 27.7i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (207.5 + 359. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-110 + 190. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 884T + 3.57e5T^{2} \) |
| 73 | \( 1 + 77T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-40 - 69.2i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (634 + 1.09e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 123T + 7.04e5T^{2} \) |
| 97 | \( 1 + (673 + 1.16e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−9.882787021758553820418328661925, −8.743189641905948510476617968059, −7.891335628204235101664423549231, −7.09990545680245645439309013406, −5.92477494287437373450261637740, −5.27255298929014226532468164052, −3.57212943905785684997026670688, −3.32790106837801027796104430513, −1.15736358828059866288076926506, −0.25975771150883997805285211717,
2.07326124701830054789114304695, 2.67092129691749386656120617319, 4.12108238436938644418309453952, 5.31489367418286294353056499834, 6.23993146961774209398349639650, 6.86857464434092634696791999615, 8.149471120742492479686463061939, 8.998023670858169098255663868691, 9.799670981182753474975626173335, 10.38416682608888887544154342341
