| L(s) = 1 | + (1.12 − 1.12i)2-s + (1.22 + 1.22i)3-s − 0.532i·4-s + 2.75·6-s + (1.65 + 1.65i)8-s + 2.99i·9-s + (0.652 − 0.652i)12-s + 4.78·16-s + (−2.10 + 2.10i)17-s + (3.37 + 3.37i)18-s − 8.60i·19-s + (−5.69 − 5.69i)23-s + 4.04i·24-s + (−3.67 + 3.67i)27-s + 11.0·31-s + (2.07 − 2.07i)32-s + ⋯ |
| L(s) = 1 | + (0.795 − 0.795i)2-s + (0.707 + 0.707i)3-s − 0.266i·4-s + 1.12·6-s + (0.583 + 0.583i)8-s + 0.999i·9-s + (0.188 − 0.188i)12-s + 1.19·16-s + (−0.510 + 0.510i)17-s + (0.795 + 0.795i)18-s − 1.97i·19-s + (−1.18 − 1.18i)23-s + 0.825i·24-s + (−0.707 + 0.707i)27-s + 1.98·31-s + (0.367 − 0.367i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.50966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.50966\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-1.12 + 1.12i)T - 2iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (2.10 - 2.10i)T - 17iT^{2} \) |
| 19 | \( 1 + 8.60iT - 19T^{2} \) |
| 23 | \( 1 + (5.69 + 5.69i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (8.28 - 8.28i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.03 + 6.03i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 + 2.42iT - 79T^{2} \) |
| 83 | \( 1 + (-12.7 - 12.7i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−11.31605015444920769247251229907, −10.65206909225237580896197523500, −9.747772739176950570505520164252, −8.626404102585444876613270472160, −7.925155186533413638504360902670, −6.47093042952993313893875088169, −4.85187421140309709850866217769, −4.35788799594172381916045016393, −3.11344107878452276496771972796, −2.23537325789500457542717505892,
1.62909029460931838809644635595, 3.35227132587942634343349705836, 4.43960524039727529739901200888, 5.84074073905846306391617541125, 6.46519756122830683122359585952, 7.59623247760244470483234184493, 8.140774535727333465925376175032, 9.501042502923627525940041571879, 10.27199559441913618208554038926, 11.80665223389160325949077047978
