| L(s) = 1 | − 2.23·2-s + (−2.23 − 2i)3-s + 1.00·4-s + (5.00 + 4.47i)6-s + 6i·7-s + 6.70·8-s + (1.00 + 8.94i)9-s + 4.47i·11-s + (−2.23 − 2.00i)12-s + 16i·13-s − 13.4i·14-s − 19·16-s + 4.47·17-s + (−2.23 − 20.0i)18-s + 2·19-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + (−0.745 − 0.666i)3-s + 0.250·4-s + (0.833 + 0.745i)6-s + 0.857i·7-s + 0.838·8-s + (0.111 + 0.993i)9-s + 0.406i·11-s + (−0.186 − 0.166i)12-s + 1.23i·13-s − 0.958i·14-s − 1.18·16-s + 0.263·17-s + (−0.124 − 1.11i)18-s + 0.105·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.316987 + 0.242157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.316987 + 0.242157i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.23 + 2i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 4T^{2} \) |
| 7 | \( 1 - 6iT - 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 - 16iT - 169T^{2} \) |
| 17 | \( 1 - 4.47T + 289T^{2} \) |
| 19 | \( 1 - 2T + 361T^{2} \) |
| 23 | \( 1 + 13.4T + 529T^{2} \) |
| 29 | \( 1 - 31.3iT - 841T^{2} \) |
| 31 | \( 1 + 18T + 961T^{2} \) |
| 37 | \( 1 - 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 49.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 4.47T + 2.80e3T^{2} \) |
| 59 | \( 1 + 4.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82T + 3.72e3T^{2} \) |
| 67 | \( 1 + 24iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 138T + 6.24e3T^{2} \) |
| 83 | \( 1 - 93.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 166iT - 9.40e3T^{2} \) |
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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
−14.49232144727837867357274384904, −13.25256448154331197846827024961, −12.10494377070694800026312630685, −11.19681487127270218773171017456, −9.918549321675346939345970827289, −8.840780556043446652023125369066, −7.66099038606770347015452052937, −6.46956883220554981972872824927, −4.87465527927019943797374885879, −1.76146658001021838191090524086,
0.56232865399743246886736453768, 3.96647598190035573314713142173, 5.55406273432819310902086409115, 7.24552569594067322114938871160, 8.465249663615244286862819829368, 9.835576344788790113193440943264, 10.42312508423581756441210393947, 11.37453480849247090349298636590, 12.88900305660941201977522951769, 14.13606309521461343941098688070
