| L(s) = 1 | + (−1.56 − 0.741i)3-s + (−3.10 − 2.60i)5-s + (−0.144 − 0.0525i)7-s + (1.90 + 2.32i)9-s + (−0.169 + 0.141i)11-s + (0.103 − 0.585i)13-s + (2.92 + 6.37i)15-s + (−2.78 + 4.81i)17-s + (1.91 + 3.30i)19-s + (0.187 + 0.189i)21-s + (−5.50 + 2.00i)23-s + (1.97 + 11.2i)25-s + (−1.25 − 5.04i)27-s + (0.129 + 0.736i)29-s + (4.77 − 1.73i)31-s + ⋯ |
| L(s) = 1 | + (−0.903 − 0.428i)3-s + (−1.38 − 1.16i)5-s + (−0.0545 − 0.0198i)7-s + (0.633 + 0.773i)9-s + (−0.0510 + 0.0428i)11-s + (0.0286 − 0.162i)13-s + (0.754 + 1.64i)15-s + (−0.674 + 1.16i)17-s + (0.438 + 0.759i)19-s + (0.0408 + 0.0413i)21-s + (−1.14 + 0.417i)23-s + (0.395 + 2.24i)25-s + (−0.241 − 0.970i)27-s + (0.0241 + 0.136i)29-s + (0.857 − 0.312i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.100719 + 0.136829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100719 + 0.136829i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.56 + 0.741i)T \) |
| good | 5 | \( 1 + (3.10 + 2.60i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (0.144 + 0.0525i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (0.169 - 0.141i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.103 + 0.585i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.78 - 4.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.91 - 3.30i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.50 - 2.00i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.129 - 0.736i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.77 + 1.73i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (1.87 - 3.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.690 + 3.91i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (7.81 - 6.56i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.447 + 0.162i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 + (5.57 + 4.67i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (3.16 + 1.15i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.29 - 7.34i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.42 + 2.47i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.638 - 1.10i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.574 - 3.25i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.43 - 8.14i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.47 - 4.29i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.33 + 3.63i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.68149488790141500028653421241, −10.75301121854014851840842422433, −9.699329154098712009124885131633, −8.229875098869069756092156345957, −8.023287788265824706987477123022, −6.76516261215334635768706710948, −5.65664020471937206818662328356, −4.64251346979757524636074629928, −3.79145687531844217354721645613, −1.46905740307732193467926497504,
0.12634927150988609280376610089, 2.89219133658631772058151191254, 4.01857178049741119854734764219, 4.90734792803220739074817549116, 6.38797910641566567999235881371, 7.00246165115584802529381193138, 7.922443530376474398743563572450, 9.218342851128069408438201215797, 10.31113231459610515049916873990, 10.95537388678154582913007445492
