| L(s) = 1 | + 3.88i·2-s − 4.12·3-s − 11.1·4-s − 16.0i·6-s − 3.44i·7-s − 27.5i·8-s + 7.98·9-s + (−6.12 + 9.13i)11-s + 45.7·12-s + 6.33i·13-s + 13.3·14-s + 62.8·16-s − 0.715i·17-s + 31.0i·18-s − 20.9i·19-s + ⋯ |
| L(s) = 1 | + 1.94i·2-s − 1.37·3-s − 2.77·4-s − 2.66i·6-s − 0.491i·7-s − 3.44i·8-s + 0.887·9-s + (−0.557 + 0.830i)11-s + 3.81·12-s + 0.487i·13-s + 0.955·14-s + 3.92·16-s − 0.0421i·17-s + 1.72i·18-s − 1.10i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.416582 + 0.222164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.416582 + 0.222164i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (6.12 - 9.13i)T \) |
| good | 2 | \( 1 - 3.88iT - 4T^{2} \) |
| 3 | \( 1 + 4.12T + 9T^{2} \) |
| 7 | \( 1 + 3.44iT - 49T^{2} \) |
| 13 | \( 1 - 6.33iT - 169T^{2} \) |
| 17 | \( 1 + 0.715iT - 289T^{2} \) |
| 19 | \( 1 + 20.9iT - 361T^{2} \) |
| 23 | \( 1 + 4.55T + 529T^{2} \) |
| 29 | \( 1 - 0.262iT - 841T^{2} \) |
| 31 | \( 1 + 9.37T + 961T^{2} \) |
| 37 | \( 1 - 19.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 24.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 63.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 34.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 60.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 47.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 108. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 96.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 20.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 98.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 114. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 127. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 3.05T + 7.92e3T^{2} \) |
| 97 | \( 1 - 91.6T + 9.40e3T^{2} \) |
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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
−12.00216518022402253116163107565, −10.68596995880395470953357543333, −9.760871592142427093592201776133, −8.690379532582876823392783419676, −7.37781101469173003560634153075, −6.90237412236070363291154398357, −5.88857369511421784244093417038, −5.02621467863297135552331282820, −4.25899995976693820439538865504, −0.39046600004020786425422682815,
0.968309033738615875176452366655, 2.64481130393154811491812606420, 3.99131110663991367100420939281, 5.29670447072257429902031097167, 5.86398110599317734447495582056, 8.019906386029781144081083092942, 9.049619431030307969043629046378, 10.25831869117257804466137889099, 10.67647664495447366256928034961, 11.56674628698638859368215372447
