| L(s) = 1 | + (1.41 − 1.91i)2-s + (2.30 − 0.435i)3-s + (−1.08 − 3.51i)4-s + (−0.708 + 2.12i)5-s + (2.42 − 5.02i)6-s + (−2.23 − 1.41i)7-s + (−3.77 − 1.32i)8-s + (2.31 − 0.908i)9-s + (3.06 + 4.36i)10-s + (−1.66 + 4.23i)11-s + (−4.02 − 7.62i)12-s + (−0.710 − 0.0800i)13-s + (−5.87 + 2.27i)14-s + (−0.707 + 5.18i)15-s + (−1.80 + 1.22i)16-s + (4.26 + 0.159i)17-s + ⋯ |
| L(s) = 1 | + (1.00 − 1.35i)2-s + (1.32 − 0.251i)3-s + (−0.542 − 1.75i)4-s + (−0.316 + 0.948i)5-s + (0.988 − 2.05i)6-s + (−0.844 − 0.535i)7-s + (−1.33 − 0.467i)8-s + (0.771 − 0.302i)9-s + (0.968 + 1.37i)10-s + (−0.501 + 1.27i)11-s + (−1.16 − 2.20i)12-s + (−0.196 − 0.0221i)13-s + (−1.57 + 0.608i)14-s + (−0.182 + 1.33i)15-s + (−0.450 + 0.307i)16-s + (1.03 + 0.0387i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63928 - 1.92124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63928 - 1.92124i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.708 - 2.12i)T \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
| good | 2 | \( 1 + (-1.41 + 1.91i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-2.30 + 0.435i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (1.66 - 4.23i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (0.710 + 0.0800i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-4.26 - 0.159i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-3.03 + 5.26i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.274 - 7.32i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (2.15 - 0.491i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (0.305 - 0.176i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.42 - 3.92i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (2.60 + 5.40i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.38 + 3.95i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-2.09 - 1.54i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (2.65 + 1.40i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.936 + 12.4i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.327 + 1.06i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-7.08 - 1.89i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.59 + 11.3i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (0.993 - 0.733i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-7.95 - 4.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.394 - 3.50i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-4.59 - 11.6i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (7.13 + 7.13i)T + 97iT^{2} \) |
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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
−12.03502764490085466120980457527, −10.95465456341203370098256084833, −9.955453010062428926046748423055, −9.503889282978965329267266557810, −7.68480911588769235897803171825, −7.02624436771930524680999535895, −5.17892788186445599527223266281, −3.63697128629338619531406238596, −3.17386769987714589853004906191, −2.06139777343382387923171036341,
3.09075735403592004653101846140, 3.85546824848697568690183844724, 5.27361784840701180417013806729, 6.06713759115376317550458056829, 7.58612362261609712847013774043, 8.308213599948199545121160589939, 8.898489119660549543683346875345, 10.06055299184574541521163380872, 12.02970058324682309978412995322, 12.76606318119121014773261966389
