| L(s) = 1 | + (0.984 − 0.173i)2-s + (0.0750 + 1.73i)3-s + (0.939 − 0.342i)4-s + (2.14 + 1.80i)5-s + (0.374 + 1.69i)6-s + (2.42 + 1.06i)7-s + (0.866 − 0.5i)8-s + (−2.98 + 0.259i)9-s + (2.42 + 1.40i)10-s + (−3.45 − 4.11i)11-s + (0.662 + 1.60i)12-s + (−2.86 − 0.505i)13-s + (2.57 + 0.626i)14-s + (−2.95 + 3.85i)15-s + (0.766 − 0.642i)16-s + (0.473 − 0.820i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (0.0433 + 0.999i)3-s + (0.469 − 0.171i)4-s + (0.960 + 0.805i)5-s + (0.152 + 0.690i)6-s + (0.915 + 0.401i)7-s + (0.306 − 0.176i)8-s + (−0.996 + 0.0865i)9-s + (0.767 + 0.443i)10-s + (−1.04 − 1.24i)11-s + (0.191 + 0.461i)12-s + (−0.794 − 0.140i)13-s + (0.687 + 0.167i)14-s + (−0.763 + 0.994i)15-s + (0.191 − 0.160i)16-s + (0.114 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04345 + 1.11169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04345 + 1.11169i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (-0.0750 - 1.73i)T \) |
| 7 | \( 1 + (-2.42 - 1.06i)T \) |
| good | 5 | \( 1 + (-2.14 - 1.80i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (3.45 + 4.11i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.86 + 0.505i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.473 + 0.820i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.211 + 0.122i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.103 - 0.283i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-6.17 + 1.08i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.27 - 3.49i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (2.84 - 4.93i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.505 + 2.86i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.36 + 6.18i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (12.6 + 4.58i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 1.40iT - 53T^{2} \) |
| 59 | \( 1 + (8.84 + 7.42i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (4.43 - 12.1i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.239 - 1.35i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (3.49 + 2.01i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.8 + 6.83i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.23 - 6.99i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.12 + 12.0i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-7.16 - 12.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.31 + 2.76i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.33219478587615606422810433218, −10.56279322851054500907936090954, −10.09314826638612874114082708154, −8.833284292456433983825401426585, −7.84022704090671592933987325973, −6.34316673810187083300511656485, −5.44143470125953270138482660309, −4.82699683936744911886680365140, −3.18216634364388678427561021776, −2.42044091449653855707688752331,
1.57893298767478166796112989285, 2.52825548955728706024763430707, 4.66478520941772046899352788041, 5.22103144930528681687059713174, 6.36313538615816459034336518224, 7.49806607146397645535489117352, 8.050389526021726011141762945684, 9.365917724907332221551136878103, 10.39692279710456962096867518782, 11.50017265432440106426934608754
