| L(s) = 1 | + (1.23 + 3.80i)2-s + (7.28 − 5.29i)3-s + (−12.9 + 9.40i)4-s + (45.5 + 32.4i)5-s + (29.1 + 21.1i)6-s − 57.2·7-s + (−51.7 − 37.6i)8-s + (25.0 − 77.0i)9-s + (−67.3 + 213. i)10-s + (6.78 + 20.8i)11-s + (−44.4 + 136. i)12-s + (−363. + 1.11e3i)13-s + (−70.7 − 217. i)14-s + (503. − 4.23i)15-s + (79.1 − 243. i)16-s + (1.04e3 + 760. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.813 + 0.580i)5-s + (0.330 + 0.239i)6-s − 0.441·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.212 + 0.674i)10-s + (0.0169 + 0.0520i)11-s + (−0.0892 + 0.274i)12-s + (−0.596 + 1.83i)13-s + (−0.0964 − 0.296i)14-s + (0.577 − 0.00486i)15-s + (0.0772 − 0.237i)16-s + (0.878 + 0.638i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.178986160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178986160\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 - 3.80i)T \) |
| 3 | \( 1 + (-7.28 + 5.29i)T \) |
| 5 | \( 1 + (-45.5 - 32.4i)T \) |
| good | 7 | \( 1 + 57.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-6.78 - 20.8i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (363. - 1.11e3i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-1.04e3 - 760. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (950. + 690. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (231. + 713. i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.50e3 - 1.09e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-5.92e3 - 4.30e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.09e3 - 9.53e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (2.84e3 - 8.76e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 8.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.75e4 - 1.27e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.89e4 + 1.37e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.30e4 + 4.00e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (2.03e3 + 6.27e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.14e4 - 8.31e3i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (2.41e4 - 1.75e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (7.24e3 + 2.22e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-6.27e4 + 4.55e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-6.21e4 - 4.51e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-3.47e4 - 1.07e5i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-5.07e4 + 3.68e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.76943412051111214393929945390, −11.65633374701017179198658926287, −10.10972275590683475091131280887, −9.348569410367634752131555141326, −8.215260270314430034694689597171, −6.78112379397992438732631735623, −6.44683317899396965135600590805, −4.80631816494810627219462031840, −3.25627334519221150792062438285, −1.82309793217181295817297655555,
0.62661903058582130575582187781, 2.30699567273646907335588331299, 3.45633266308720874453544487095, 4.99746358730999844775562751515, 5.88448052882173355487881839033, 7.75787724091627880688759278477, 8.879909301370121709065838710143, 10.00743013688748899884486316452, 10.28027886847271861874038183407, 11.93319319816757528956307769459
