L(s) = 1 | + (−3.55 − 2.05i)7-s + (−1.90 + 3.30i)11-s + (−5.03 + 2.90i)13-s − 3.81i·17-s + 1.81·19-s + (1.81 − 1.05i)23-s + (3.60 − 6.23i)29-s + (0.908 + 1.57i)31-s − 6.01i·37-s + (5.50 + 9.54i)41-s + (5.03 + 2.90i)43-s + (10.3 + 5.95i)47-s + (4.90 + 8.50i)49-s + 4.20i·53-s + (2.10 + 3.63i)59-s + ⋯ |
L(s) = 1 | + (−1.34 − 0.774i)7-s + (−0.575 + 0.996i)11-s + (−1.39 + 0.806i)13-s − 0.925i·17-s + 0.416·19-s + (0.379 − 0.219i)23-s + (0.668 − 1.15i)29-s + (0.163 + 0.282i)31-s − 0.989i·37-s + (0.860 + 1.49i)41-s + (0.768 + 0.443i)43-s + (1.50 + 0.869i)47-s + (0.701 + 1.21i)49-s + 0.577i·53-s + (0.273 + 0.473i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.125342716\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.125342716\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3.55 + 2.05i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.90 - 3.30i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.03 - 2.90i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.81iT - 17T^{2} \) |
| 19 | \( 1 - 1.81T + 19T^{2} \) |
| 23 | \( 1 + (-1.81 + 1.05i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.60 + 6.23i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.908 - 1.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.01iT - 37T^{2} \) |
| 41 | \( 1 + (-5.50 - 9.54i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.03 - 2.90i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 5.95i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 4.20iT - 53T^{2} \) |
| 59 | \( 1 + (-2.10 - 3.63i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.50 + 2.61i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.21 + 1.85i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.01T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.37 - 1.94i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (10.5 + 6.10i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.212986403260496757501807818766, −7.77296318854951789321142876893, −7.29368904203461010062281877228, −6.75798023742244015200624865672, −5.84269590325525535776881596796, −4.69082959481514185017369377389, −4.27778040150022523127257265239, −2.96524951376431099489340924249, −2.37038790927892182735610537423, −0.65796594437379321937397215146,
0.60537912869646132882651209569, 2.43191589161401916961304338011, 3.00051623207357923805341987405, 3.84963856069571611652511907182, 5.30049100401084539047387412339, 5.57234593193224085799970573424, 6.52571804656264477625674873283, 7.26765152833148764426121222278, 8.159946361328233054653343377000, 8.861624776914030331625390168955