| 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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.633237633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.633237633\) |
| \(L(2)\) |
|
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
−9.154071031578695208127431266773, −8.586494948505110807863364777983, −7.45453613036957750764571838111, −6.64759357167885088321465986761, −5.85772951533293728225631844988, −5.16960599978193946689212058817, −4.33305602213157078972375944572, −2.57058222591157623448371248804, −2.45120203278259882416029388286, −0.876286958305806053852312984119,
0.956968111036679285865570597679, 2.02849608136922062627983733643, 3.01154351758696967411052925135, 4.20197831780928977442731535222, 5.15018422750699677646040680163, 5.91021305358986767979492038948, 6.65171500439766820606778109934, 7.55210532330969928741734268764, 8.304473236900660885379032180258, 9.397192408380223630135869807556
