| L(s) = 1 | − 2.23·2-s + 1.00·4-s − 3i·7-s + 6.70·8-s + 4.47i·11-s − 14i·13-s + 6.70i·14-s − 19·16-s − 8.94·17-s − 19-s − 10.0i·22-s + 40.2·23-s + 31.3i·26-s − 3.00i·28-s + 44.7i·29-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.250·4-s − 0.428i·7-s + 0.838·8-s + 0.406i·11-s − 1.07i·13-s + 0.479i·14-s − 1.18·16-s − 0.526·17-s − 0.0526·19-s − 0.454i·22-s + 1.74·23-s + 1.20i·26-s − 0.107i·28-s + 1.54i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5354314758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5354314758\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 4T^{2} \) |
| 7 | \( 1 + 3iT - 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 + 14iT - 169T^{2} \) |
| 17 | \( 1 + 8.94T + 289T^{2} \) |
| 19 | \( 1 + T + 361T^{2} \) |
| 23 | \( 1 - 40.2T + 529T^{2} \) |
| 29 | \( 1 - 44.7iT - 841T^{2} \) |
| 31 | \( 1 + 21T + 961T^{2} \) |
| 37 | \( 1 + 53iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 22.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 76.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 49.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 84.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 71T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 58.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 51T + 6.24e3T^{2} \) |
| 83 | \( 1 + 134.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 120. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 143iT - 9.40e3T^{2} \) |
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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
−9.963781488077124219403721699070, −9.085279107513740379648597204737, −8.481121867174354650032497262524, −7.41310226734447314945736181534, −6.96090584944528135180069372433, −5.43370362690382297935395744762, −4.51007104219573635031006154222, −3.16563465753486925403996131259, −1.59980605476306036530939602920, −0.32331718035457609077946206335,
1.23483731109388377972496096318, 2.54345925457337126023091905840, 4.11235597463626588751793704386, 5.09642793100807891330049922531, 6.40569802439083701760613598984, 7.22228170188870827875721366486, 8.233864228478282170975467505841, 8.959642230718643480936718714540, 9.454649899251677730009421280660, 10.43354428839010481000865190805
