| L(s) = 1 | + 1.74i·3-s − 6.55·7-s + 5.94·9-s − 5.76i·11-s − 19.9i·13-s + 21.9i·17-s − 27.2·19-s − 11.4i·21-s + 3.18i·23-s + 26.1i·27-s − 13.6i·29-s + (−22.6 + 21.2i)31-s + 10.0·33-s − 11.9i·37-s + 34.8·39-s + ⋯ |
| L(s) = 1 | + 0.582i·3-s − 0.935·7-s + 0.660·9-s − 0.524i·11-s − 1.53i·13-s + 1.29i·17-s − 1.43·19-s − 0.544i·21-s + 0.138i·23-s + 0.967i·27-s − 0.470i·29-s + (−0.729 + 0.683i)31-s + 0.305·33-s − 0.323i·37-s + 0.892·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.565969372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565969372\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + (22.6 - 21.2i)T \) |
| good | 3 | \( 1 - 1.74iT - 9T^{2} \) |
| 7 | \( 1 + 6.55T + 49T^{2} \) |
| 11 | \( 1 + 5.76iT - 121T^{2} \) |
| 13 | \( 1 + 19.9iT - 169T^{2} \) |
| 17 | \( 1 - 21.9iT - 289T^{2} \) |
| 19 | \( 1 + 27.2T + 361T^{2} \) |
| 23 | \( 1 - 3.18iT - 529T^{2} \) |
| 29 | \( 1 + 13.6iT - 841T^{2} \) |
| 37 | \( 1 + 11.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 15.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 46.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 77.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 32.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 24.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 92.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 95.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 7.07iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 55.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 117.T + 9.40e3T^{2} \) |
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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
−8.579896485830185143828045797806, −8.024877071948318754040134816677, −7.05531671480489096399795907862, −6.22482943989847755710472877284, −5.67922922253693775984091725134, −4.66356863701660668404429884241, −3.75216157931414210577727685102, −3.27341567890951008792174397925, −2.05544447579321591677007863382, −0.67652069083781041837458013415,
0.52597979590601581609428495439, 1.86408932405572448490024776731, 2.48875507325873293115550098262, 3.84171918341778204803093894051, 4.38856400100161655712671462179, 5.42424186141645253379947878465, 6.58729115271995198771125266325, 6.81411052212797521865575776261, 7.40992639952696482457627592773, 8.490781030439557448862976050951
