L(s) = 1 | + (2.23 − 2i)3-s + 2i·7-s + (1.00 − 8.94i)9-s + 13.4i·11-s + 8i·13-s − 13.4·17-s − 34·19-s + (4 + 4.47i)21-s − 40.2·23-s + (−15.6 − 22.0i)27-s − 40.2i·29-s − 14·31-s + (26.8 + 30.0i)33-s − 56i·37-s + (16 + 17.8i)39-s + ⋯ |
L(s) = 1 | + (0.745 − 0.666i)3-s + 0.285i·7-s + (0.111 − 0.993i)9-s + 1.21i·11-s + 0.615i·13-s − 0.789·17-s − 1.78·19-s + (0.190 + 0.212i)21-s − 1.74·23-s + (−0.579 − 0.814i)27-s − 1.38i·29-s − 0.451·31-s + (0.813 + 0.909i)33-s − 1.51i·37-s + (0.410 + 0.458i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01056876983\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01056876983\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.23 + 2i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2iT - 49T^{2} \) |
| 11 | \( 1 - 13.4iT - 121T^{2} \) |
| 13 | \( 1 - 8iT - 169T^{2} \) |
| 17 | \( 1 + 13.4T + 289T^{2} \) |
| 19 | \( 1 + 34T + 361T^{2} \) |
| 23 | \( 1 + 40.2T + 529T^{2} \) |
| 29 | \( 1 + 40.2iT - 841T^{2} \) |
| 31 | \( 1 + 14T + 961T^{2} \) |
| 37 | \( 1 + 56iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 40.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 13.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 46T + 3.72e3T^{2} \) |
| 67 | \( 1 - 32iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 53.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 22T + 6.24e3T^{2} \) |
| 83 | \( 1 - 120.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 122iT - 9.40e3T^{2} \) |
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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
−9.079774559481353912208652494311, −8.233002992134917170301318635293, −7.54683543673503006692789183266, −6.61146285672673865505168395977, −6.05349648417671664937376774984, −4.48244269934000413310430976965, −3.91128773101413937118617238929, −2.22765364733967551678613987539, −2.00793518363584693449368993657, −0.00247494265833574726372291768,
1.86220032734003957367456191315, 3.01370392120000591473021930520, 3.86263442570371877847876657115, 4.69707560066858465016458093595, 5.79720231659006233478279235949, 6.66638928993202976700456397516, 7.88602232359321335209910843962, 8.456007467533345531558320566434, 9.027989872721636658977031575240, 10.12157897081506421695098477130