| L(s) = 1 | + 3.79·3-s − 2.23i·5-s − 7.10i·7-s + 5.39·9-s − 11.2i·11-s + 20.0·13-s − 8.48i·15-s + 1.63i·17-s − 29.4i·19-s − 26.9i·21-s + (−20.0 − 11.3i)23-s − 5.00·25-s − 13.6·27-s − 50.3·29-s − 11.1·31-s + ⋯ |
| L(s) = 1 | + 1.26·3-s − 0.447i·5-s − 1.01i·7-s + 0.599·9-s − 1.02i·11-s + 1.54·13-s − 0.565i·15-s + 0.0959i·17-s − 1.54i·19-s − 1.28i·21-s + (−0.870 − 0.491i)23-s − 0.200·25-s − 0.506·27-s − 1.73·29-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.760728667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.760728667\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (20.0 + 11.3i)T \) |
| good | 3 | \( 1 - 3.79T + 9T^{2} \) |
| 7 | \( 1 + 7.10iT - 49T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 - 20.0T + 169T^{2} \) |
| 17 | \( 1 - 1.63iT - 289T^{2} \) |
| 19 | \( 1 + 29.4iT - 361T^{2} \) |
| 29 | \( 1 + 50.3T + 841T^{2} \) |
| 31 | \( 1 + 11.1T + 961T^{2} \) |
| 37 | \( 1 - 40.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 7.24T + 1.68e3T^{2} \) |
| 43 | \( 1 - 71.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 6.40T + 2.20e3T^{2} \) |
| 53 | \( 1 + 20.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 37.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 124. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 43.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 48.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 9.63iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143. iT - 9.40e3T^{2} \) |
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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
−8.705347463310267390526088745437, −8.222787838119423295214572305986, −7.46440134835743698813259902377, −6.49798707213312731468432298801, −5.62693258085392190259632657567, −4.35363524130945903526966092885, −3.68829246816931302261031874506, −2.95990859229579918707980204231, −1.66748380117080390098607543823, −0.56167408804772714229406656280,
1.81153596413117049310131877464, 2.24770826565601784403130164663, 3.64609199629847666346034425723, 3.80596083873942597897100596692, 5.52174010951238279135670929389, 6.00189905967125364739433722809, 7.22470845461868656910157343592, 7.86189712849230918551304455018, 8.602350344829600497412957609196, 9.190934635853869354158396987936
