| L(s) = 1 | + (−0.997 + 0.0713i)2-s + (0.212 − 0.977i)3-s + (0.989 − 0.142i)4-s + (2.23 + 0.0191i)5-s + (−0.142 + 0.989i)6-s + (0.197 − 0.362i)7-s + (−0.977 + 0.212i)8-s + (−0.909 − 0.415i)9-s + (−2.23 + 0.140i)10-s + (2.14 − 1.85i)11-s + (0.0713 − 0.997i)12-s + (1.42 − 0.776i)13-s + (−0.171 + 0.375i)14-s + (0.494 − 2.18i)15-s + (0.959 − 0.281i)16-s + (−0.178 − 0.238i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 + 0.0504i)2-s + (0.122 − 0.564i)3-s + (0.494 − 0.0711i)4-s + (0.999 + 0.00857i)5-s + (−0.0580 + 0.404i)6-s + (0.0747 − 0.136i)7-s + (−0.345 + 0.0751i)8-s + (−0.303 − 0.138i)9-s + (−0.705 + 0.0443i)10-s + (0.646 − 0.560i)11-s + (0.0205 − 0.287i)12-s + (0.394 − 0.215i)13-s + (−0.0458 + 0.100i)14-s + (0.127 − 0.563i)15-s + (0.239 − 0.0704i)16-s + (−0.0432 − 0.0578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25751 - 0.628376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25751 - 0.628376i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.997 - 0.0713i)T \) |
| 3 | \( 1 + (-0.212 + 0.977i)T \) |
| 5 | \( 1 + (-2.23 - 0.0191i)T \) |
| 23 | \( 1 + (0.593 + 4.75i)T \) |
| good | 7 | \( 1 + (-0.197 + 0.362i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-2.14 + 1.85i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.42 + 0.776i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (0.178 + 0.238i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.324 - 2.25i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (2.51 + 0.361i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.05 - 3.24i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.563 - 1.50i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-0.810 - 1.77i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (2.53 + 0.551i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (8.55 + 8.55i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.215 + 0.117i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-2.89 + 9.84i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.71 + 4.21i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.678 + 9.48i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-0.0645 + 0.0745i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.73 - 2.04i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-8.99 - 2.64i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-11.4 + 4.27i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (2.18 - 1.40i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (8.47 + 3.16i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.24812240891592671179244141929, −9.419146428118666901789807305093, −8.602401061961406320305430072013, −7.930069538666688190168224425924, −6.61804600771268829107490536139, −6.29459733984700667651725345236, −5.12235305100703070006552677991, −3.44661816473388323720023474514, −2.18697310295087341768877314622, −1.05030217276313763633429662897,
1.50311770458340546376064679373, 2.67999851327686148806020099380, 4.04092019071279982905292950399, 5.26653779032371070843142215678, 6.20830122516539106911173300844, 7.09162012749884673983700186555, 8.244401535189556583288929861216, 9.192009305537638098349724102626, 9.572580816089701531037794520760, 10.36578927465203615046845706699
