| L(s) = 1 | + (−0.997 + 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (−1.89 − 1.18i)5-s + (0.142 − 0.989i)6-s + (0.244 − 0.448i)7-s + (−0.977 + 0.212i)8-s + (−0.909 − 0.415i)9-s + (1.97 + 1.04i)10-s + (−1.95 + 1.69i)11-s + (−0.0713 + 0.997i)12-s + (3.92 − 2.14i)13-s + (−0.212 + 0.464i)14-s + (1.56 − 1.59i)15-s + (0.959 − 0.281i)16-s + (−0.573 − 0.765i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 + 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (−0.847 − 0.531i)5-s + (0.0580 − 0.404i)6-s + (0.0925 − 0.169i)7-s + (−0.345 + 0.0751i)8-s + (−0.303 − 0.138i)9-s + (0.624 + 0.331i)10-s + (−0.589 + 0.510i)11-s + (−0.0205 + 0.287i)12-s + (1.08 − 0.594i)13-s + (−0.0566 + 0.124i)14-s + (0.403 − 0.412i)15-s + (0.239 − 0.0704i)16-s + (−0.138 − 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.590281 + 0.463945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.590281 + 0.463945i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.89 + 1.18i)T \) |
| 23 | \( 1 + (0.985 - 4.69i)T \) |
| good | 7 | \( 1 + (-0.244 + 0.448i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (1.95 - 1.69i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.92 + 2.14i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (0.573 + 0.765i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.787 - 5.47i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-9.80 - 1.40i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (1.87 + 1.20i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.327 - 0.879i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-3.97 - 8.70i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (0.920 + 0.200i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (5.32 + 5.32i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.72 + 1.49i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (3.62 - 12.3i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (2.08 - 3.24i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.439 - 6.14i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (5.17 - 5.97i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.38 - 1.78i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-13.7 - 4.02i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-14.0 + 5.24i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-1.23 + 0.791i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (6.31 + 2.35i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.51133467041589680545390941417, −9.875833014680858039395084867163, −8.844283937869531133752398004197, −8.110241480395541604841472846825, −7.52316649763920065950771227885, −6.20842692864436382536466833502, −5.20349107776014850991485625315, −4.13455564415910039685399077735, −3.10010193479229447172048451112, −1.19969757642523337111893823456,
0.59423675798861870788109995811, 2.33316217498783040983758978648, 3.40523104481001619845833507482, 4.77868569292086667978967407426, 6.28161877062730087469190609825, 6.75770768875782096461929369386, 7.85294763557638030807088494417, 8.416122571494316464997544648444, 9.207389092526685131912167870709, 10.71874603444476223631062912924
