| L(s) = 1 | + 1.41·2-s − 3.79·3-s + 2.00·4-s + 2.23i·5-s − 5.36·6-s − 7.10i·7-s + 2.82·8-s + 5.39·9-s + 3.16i·10-s − 11.2i·11-s − 7.58·12-s + 20.0·13-s − 10.0i·14-s − 8.48i·15-s + 4.00·16-s − 1.63i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.26·3-s + 0.500·4-s + 0.447i·5-s − 0.894·6-s − 1.01i·7-s + 0.353·8-s + 0.599·9-s + 0.316i·10-s − 1.02i·11-s − 0.632·12-s + 1.54·13-s − 0.717i·14-s − 0.565i·15-s + 0.250·16-s − 0.0959i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.29426 - 0.755548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29426 - 0.755548i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 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
−11.33918849336548631448953180922, −11.20798132866916620441703950772, −10.52988249849837233720093652202, −8.887033016110281098810632837438, −7.36402831250250469632618451378, −6.44042405350931968204494307831, −5.70863552444860174755606294982, −4.45438297097648479535958495247, −3.23847693274474997776256629353, −0.815448808489126174446673234449,
1.66391035317219180080877311385, 3.72284431589068406072665283529, 5.12066023579721801194227320886, 5.74106479173759410320965801997, 6.61874850286461147283018768220, 8.088052256246064828287763028984, 9.320764904577666268190035336662, 10.58217597153176519776060567316, 11.43760336526825084812491635532, 12.19560258564924662308479708192
