| L(s) = 1 | − 1.73i·3-s + 2.23·5-s − 4.28i·7-s − 2.99·9-s − 11.2i·11-s + 5.41·13-s − 3.87i·15-s − 1.41·17-s − 8.56i·19-s − 7.41·21-s − 24.0i·23-s + 5.00·25-s + 5.19i·27-s − 25.4·29-s − 50.1i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447·5-s − 0.611i·7-s − 0.333·9-s − 1.01i·11-s + 0.416·13-s − 0.258i·15-s − 0.0833·17-s − 0.450i·19-s − 0.353·21-s − 1.04i·23-s + 0.200·25-s + 0.192i·27-s − 0.876·29-s − 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09462 - 1.09462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09462 - 1.09462i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 + 4.28iT - 49T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 - 5.41T + 169T^{2} \) |
| 17 | \( 1 + 1.41T + 289T^{2} \) |
| 19 | \( 1 + 8.56iT - 361T^{2} \) |
| 23 | \( 1 + 24.0iT - 529T^{2} \) |
| 29 | \( 1 + 25.4T + 841T^{2} \) |
| 31 | \( 1 + 50.1iT - 961T^{2} \) |
| 37 | \( 1 - 60.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30T + 1.68e3T^{2} \) |
| 43 | \( 1 - 37.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 39.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 16.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 81.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 94.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 129. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 24.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 91.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 88.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 20.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 14T + 9.40e3T^{2} \) |
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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
−11.50328640928012969513740952567, −10.89851490351746656181049476025, −9.718628388953054439633031973265, −8.644515978466745917765062884235, −7.69382418144301866457884565608, −6.53511652046397327337993089806, −5.70426983897486499443157401130, −4.13303555463671937849870148844, −2.60834843336393715955041397678, −0.859092968332865621591703998275,
1.95610550847262223300549683189, 3.53208496918929497047413446254, 4.92384509019044727649073198166, 5.85049570954715614686803995000, 7.10882758471019661522553600375, 8.415162715502546197641717428944, 9.388332930598228828812567627812, 10.07419371826540934968148116668, 11.14121732527518704501894494978, 12.11465377786315986319264747588
