L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.01 − 1.40i)3-s − 1.00i·4-s + (−2.22 + 0.198i)5-s + (−1.71 − 0.275i)6-s + (0.787 + 0.787i)7-s + (−0.707 − 0.707i)8-s + (−0.940 + 2.84i)9-s + (−1.43 + 1.71i)10-s + 0.307i·11-s + (−1.40 + 1.01i)12-s + (−4.92 + 4.92i)13-s + 1.11·14-s + (2.53 + 2.92i)15-s − 1.00·16-s + (−0.790 + 0.790i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.585 − 0.810i)3-s − 0.500i·4-s + (−0.996 + 0.0889i)5-s + (−0.698 − 0.112i)6-s + (0.297 + 0.297i)7-s + (−0.250 − 0.250i)8-s + (−0.313 + 0.949i)9-s + (−0.453 + 0.542i)10-s + 0.0926i·11-s + (−0.405 + 0.292i)12-s + (−1.36 + 1.36i)13-s + 0.297·14-s + (0.655 + 0.755i)15-s − 0.250·16-s + (−0.191 + 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.356876 + 0.262536i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.356876 + 0.262536i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.01 + 1.40i)T \) |
| 5 | \( 1 + (2.22 - 0.198i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-0.787 - 0.787i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.307iT - 11T^{2} \) |
| 13 | \( 1 + (4.92 - 4.92i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.790 - 0.790i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.691iT - 19T^{2} \) |
| 29 | \( 1 + 6.67T + 29T^{2} \) |
| 31 | \( 1 - 4.52T + 31T^{2} \) |
| 37 | \( 1 + (-1.77 - 1.77i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.596iT - 41T^{2} \) |
| 43 | \( 1 + (6.70 - 6.70i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.88 - 6.88i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.94 - 5.94i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.38T + 59T^{2} \) |
| 61 | \( 1 + 9.95T + 61T^{2} \) |
| 67 | \( 1 + (-2.19 - 2.19i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.55iT - 71T^{2} \) |
| 73 | \( 1 + (4.90 - 4.90i)T - 73iT^{2} \) |
| 79 | \( 1 + 15.7iT - 79T^{2} \) |
| 83 | \( 1 + (6.65 + 6.65i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + (12.8 + 12.8i)T + 97iT^{2} \) |
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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
−11.11801595343191502230032859565, −9.978413879802863393692690741657, −8.909497648238718855460885014158, −7.80235556678971634043893581091, −7.09437562834892406455996151205, −6.23579875106732543585745490379, −4.96021973744224511832123185674, −4.38044534831117413900461081468, −2.84428990477654759009824837332, −1.67555766279136248179195508410,
0.20868268522445718704644295014, 3.00366421636742135127644288676, 3.96590567592336470912276242550, 4.88606881363354713139785810386, 5.46601242834962373922395322824, 6.75825205322583126583583810020, 7.61638246538239531820760045221, 8.362397070788364048645281733084, 9.500422395457275007594101226418, 10.43642634084845387041710205179