L(s) = 1 | + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (1.48 − 4.77i)5-s + 2.44·6-s + (−9.54 − 9.54i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−6.26 + 3.28i)10-s − 15.2·11-s + (−2.44 − 2.44i)12-s + (−8.82 + 8.82i)13-s + 19.0i·14-s + (4.02 + 7.66i)15-s − 4·16-s + (5.29 + 5.29i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (0.297 − 0.954i)5-s + 0.408·6-s + (−1.36 − 1.36i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.626 + 0.328i)10-s − 1.38·11-s + (−0.204 − 0.204i)12-s + (−0.678 + 0.678i)13-s + 1.36i·14-s + (0.268 + 0.511i)15-s − 0.250·16-s + (0.311 + 0.311i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2589028396\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2589028396\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + (-1.48 + 4.77i)T \) |
| 23 | \( 1 + (-3.39 + 3.39i)T \) |
good | 7 | \( 1 + (9.54 + 9.54i)T + 49iT^{2} \) |
| 11 | \( 1 + 15.2T + 121T^{2} \) |
| 13 | \( 1 + (8.82 - 8.82i)T - 169iT^{2} \) |
| 17 | \( 1 + (-5.29 - 5.29i)T + 289iT^{2} \) |
| 19 | \( 1 + 0.807iT - 361T^{2} \) |
| 29 | \( 1 - 14.0iT - 841T^{2} \) |
| 31 | \( 1 - 29.6T + 961T^{2} \) |
| 37 | \( 1 + (6.74 + 6.74i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 70.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36.3 + 36.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (32.5 + 32.5i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (41.7 - 41.7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 103. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 63.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (31.8 + 31.8i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 61.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-80.7 + 80.7i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 132. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (59.6 - 59.6i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 87.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-71.4 - 71.4i)T + 9.40e3iT^{2} \) |
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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
−10.28061030551962504553157302004, −9.748578373834742293303170674478, −9.017615747446551776339038225212, −7.83597716426055059701565657655, −7.02342263946279135182954276941, −5.90135654561546059169969954723, −4.73616267855345823366043762072, −3.91464008459052918497381587499, −2.65069273198130671056461268452, −0.889760812329840278619189493661,
0.14763301494521697769852231252, 2.43955324293379676695734630844, 2.96842235425101773891035979722, 5.15675454555104635229721284492, 5.88509603801742699342045341455, 6.47963631270921880638421663373, 7.47586998472261079047778124329, 8.184266384551577118878021408810, 9.535310548724715501087251630792, 9.896527350659777396104948332116