| L(s) = 1 | − 6.59·5-s − 4.56i·7-s + 1.16i·11-s − 13.5·13-s − 5.32·17-s + 25.1i·19-s − 10.4i·23-s + 18.5·25-s − 47.0·29-s + 22.3i·31-s + 30.0i·35-s + 60.7·37-s − 8.81·41-s − 29.1i·43-s − 78.2i·47-s + ⋯ |
| L(s) = 1 | − 1.31·5-s − 0.651i·7-s + 0.105i·11-s − 1.04·13-s − 0.313·17-s + 1.32i·19-s − 0.453i·23-s + 0.740·25-s − 1.62·29-s + 0.722i·31-s + 0.859i·35-s + 1.64·37-s − 0.215·41-s − 0.677i·43-s − 1.66i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9394602411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9394602411\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6.59T + 25T^{2} \) |
| 7 | \( 1 + 4.56iT - 49T^{2} \) |
| 11 | \( 1 - 1.16iT - 121T^{2} \) |
| 13 | \( 1 + 13.5T + 169T^{2} \) |
| 17 | \( 1 + 5.32T + 289T^{2} \) |
| 19 | \( 1 - 25.1iT - 361T^{2} \) |
| 23 | \( 1 + 10.4iT - 529T^{2} \) |
| 29 | \( 1 + 47.0T + 841T^{2} \) |
| 31 | \( 1 - 22.3iT - 961T^{2} \) |
| 37 | \( 1 - 60.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.81T + 1.68e3T^{2} \) |
| 43 | \( 1 + 29.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 78.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 109. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 66.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 81.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 4.29T + 5.32e3T^{2} \) |
| 79 | \( 1 - 5.20iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 141.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 136.T + 9.40e3T^{2} \) |
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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
−9.069430539420396299631487956388, −8.163591482699218626640506970971, −7.51666954418492289452302551733, −7.07198512941465851506439228182, −5.91489222572362610782965094887, −4.83107629281267379147351912232, −4.05904381212016117248343016257, −3.41084347102244334703632373978, −2.05693874097741981822762830370, −0.53837754499369090984439100671,
0.49100809763707120611175436827, 2.22843647115315784135964639149, 3.14275218271398358229483798098, 4.20150719954522756017223465779, 4.88674000230884909754615753262, 5.88011682402177885960820072648, 6.92353742733050492863843233033, 7.65917488767265795650734408317, 8.159161522049558678669149987134, 9.294434067716921733135295202029
