| L(s) = 1 | + (−0.173 + 0.984i)2-s + (−1.49 − 0.879i)3-s + (−0.939 − 0.342i)4-s + (−2.11 + 1.77i)5-s + (1.12 − 1.31i)6-s + (0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (1.45 + 2.62i)9-s + (−1.37 − 2.38i)10-s + (−0.597 − 0.501i)11-s + (1.10 + 1.33i)12-s + (−0.677 − 3.84i)13-s + (0.173 + 0.984i)14-s + (4.71 − 0.787i)15-s + (0.766 + 0.642i)16-s + (−2.34 − 4.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.861 − 0.507i)3-s + (−0.469 − 0.171i)4-s + (−0.945 + 0.793i)5-s + (0.459 − 0.537i)6-s + (0.355 − 0.129i)7-s + (0.176 − 0.306i)8-s + (0.484 + 0.874i)9-s + (−0.436 − 0.755i)10-s + (−0.180 − 0.151i)11-s + (0.317 + 0.385i)12-s + (−0.187 − 1.06i)13-s + (0.0464 + 0.263i)14-s + (1.21 − 0.203i)15-s + (0.191 + 0.160i)16-s + (−0.569 − 0.986i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.494527 - 0.256772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494527 - 0.256772i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (1.49 + 0.879i)T \) |
| 7 | \( 1 + (-0.939 + 0.342i)T \) |
| good | 5 | \( 1 + (2.11 - 1.77i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (0.597 + 0.501i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.677 + 3.84i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.34 + 4.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.82 + 6.63i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.02 - 1.10i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.791 + 4.49i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.85 - 0.676i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (4.95 + 8.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.497 - 2.81i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.61 - 2.19i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (7.88 - 2.86i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 + (-2.53 + 2.12i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (10.1 - 3.70i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.20 - 6.81i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.570 + 0.987i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.89 + 13.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.445 + 2.52i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.26 + 12.8i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (5.93 - 10.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.85 + 7.42i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.21222001425039228587100654797, −10.60582696144380069449299538242, −9.318009496171977781845898264776, −7.944248639515260558056433938942, −7.38114955473922500600776949728, −6.71825942679860600144284250485, −5.44450648067804598051829527832, −4.60655199862079113052097423347, −2.93901606946126637642896936569, −0.47536827634126041880444416736,
1.43887369794951575400281618214, 3.65059379265008102334591374697, 4.47822257505364093849301328582, 5.30054562569413212768630154176, 6.71735456092515486602875273181, 8.047926178901805308261711358410, 8.852340925705477579470167662826, 9.848361977054387773620540540091, 10.74051902390974855965804219529, 11.59268186455678300962673367556
