| L(s) = 1 | + 7.77i·5-s + 13.6·7-s + 2.59i·11-s + 8.76·13-s + 17.6i·17-s − 2.93·19-s − 18.5i·23-s − 35.3·25-s − 3.09i·29-s + 40.0·31-s + 106. i·35-s + 25.6·37-s + 62.1i·41-s + 32.9·43-s + 52.0i·47-s + ⋯ |
| L(s) = 1 | + 1.55i·5-s + 1.95·7-s + 0.235i·11-s + 0.674·13-s + 1.03i·17-s − 0.154·19-s − 0.804i·23-s − 1.41·25-s − 0.106i·29-s + 1.29·31-s + 3.03i·35-s + 0.692·37-s + 1.51i·41-s + 0.765·43-s + 1.10i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.029963542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.029963542\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 7.77iT - 25T^{2} \) |
| 7 | \( 1 - 13.6T + 49T^{2} \) |
| 11 | \( 1 - 2.59iT - 121T^{2} \) |
| 13 | \( 1 - 8.76T + 169T^{2} \) |
| 17 | \( 1 - 17.6iT - 289T^{2} \) |
| 19 | \( 1 + 2.93T + 361T^{2} \) |
| 23 | \( 1 + 18.5iT - 529T^{2} \) |
| 29 | \( 1 + 3.09iT - 841T^{2} \) |
| 31 | \( 1 - 40.0T + 961T^{2} \) |
| 37 | \( 1 - 25.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 32.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 52.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 53.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 103. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.95T + 3.72e3T^{2} \) |
| 67 | \( 1 - 47.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 89.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 67.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 36.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 20.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 96.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 67.2T + 9.40e3T^{2} \) |
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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
−8.441401211704989034024563616748, −8.047190871782692930921638476915, −7.38255533806090809740529185589, −6.40935505408281119371780831389, −5.95106194187243947364242223023, −4.70538445482922417677527702859, −4.17358134411921408955572051578, −3.01194301815471384839878823838, −2.16534673851084021021377950604, −1.24304695629968751633089665944,
0.78216863920009119470387727296, 1.37196527013028419454215386893, 2.39615465447384360661663046191, 3.95652125015680860712041600787, 4.56037189310425757985678244880, 5.28773846398334513215237042577, 5.69794884016493624002530227401, 7.10573226431782409816732821482, 7.85978805331569432436697342437, 8.528559815295258387906389272339
