| L(s) = 1 | + (−0.707 − 1.87i)2-s + (−3 + 2.64i)4-s + 1.54i·5-s − 2.64i·7-s + (7.07 + 3.74i)8-s + (2.89 − 1.09i)10-s + 4.48·11-s + 1.54i·13-s + (−4.94 + 1.87i)14-s + (1.99 − 15.8i)16-s − 23.6·17-s − 24.8·19-s + (−4.10 − 4.64i)20-s + (−3.17 − 8.39i)22-s + 35.2i·23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.935i)2-s + (−0.750 + 0.661i)4-s + 0.309i·5-s − 0.377i·7-s + (0.883 + 0.467i)8-s + (0.289 − 0.109i)10-s + 0.407·11-s + 0.119i·13-s + (−0.353 + 0.133i)14-s + (0.124 − 0.992i)16-s − 1.39·17-s − 1.30·19-s + (−0.205 − 0.232i)20-s + (−0.144 − 0.381i)22-s + 1.53i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7156956089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7156956089\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 + 1.87i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 - 1.54iT - 25T^{2} \) |
| 11 | \( 1 - 4.48T + 121T^{2} \) |
| 13 | \( 1 - 1.54iT - 169T^{2} \) |
| 17 | \( 1 + 23.6T + 289T^{2} \) |
| 19 | \( 1 + 24.8T + 361T^{2} \) |
| 23 | \( 1 - 35.2iT - 529T^{2} \) |
| 29 | \( 1 + 22.4iT - 841T^{2} \) |
| 31 | \( 1 - 46.7iT - 961T^{2} \) |
| 37 | \( 1 - 58.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 17.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 36.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 97.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 61.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 37.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 33.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 69.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 3.61T + 6.88e3T^{2} \) |
| 89 | \( 1 + 44.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 96.1T + 9.40e3T^{2} \) |
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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
−10.85843073848488798705503576270, −10.16047030317582775408085609020, −9.128090830965024747747021751895, −8.487474802287152374822615221959, −7.32697364304535203234088175289, −6.41367046605794350634538784193, −4.81640815089583845139797447632, −3.94575470737765481742716761213, −2.75792904433651014084865984477, −1.45385862046672583426022986894,
0.33144843050041846784517486301, 2.15559199385678359314322573754, 4.13807116486879062348051106538, 4.91864477720014140632659028792, 6.18995866995283941778300219265, 6.72370846111038681277716964338, 7.955031717038253761804793249984, 8.815452666860552159909321060670, 9.236804227725063761090674367391, 10.51195736285197926984333145260
