| L(s) = 1 | + (0.246 + 1.39i)2-s + (0.290 + 0.956i)3-s + (−1.87 + 0.686i)4-s + (0.267 + 2.71i)5-s + (−1.26 + 0.640i)6-s + (−0.343 + 0.513i)7-s + (−1.41 − 2.44i)8-s + (−0.831 + 0.555i)9-s + (−3.71 + 1.04i)10-s + (0.294 − 0.157i)11-s + (−1.20 − 1.59i)12-s + (−2.51 − 0.247i)13-s + (−0.800 − 0.351i)14-s + (−2.52 + 1.04i)15-s + (3.05 − 2.57i)16-s + (4.21 + 1.74i)17-s + ⋯ |
| L(s) = 1 | + (0.174 + 0.984i)2-s + (0.167 + 0.552i)3-s + (−0.939 + 0.343i)4-s + (0.119 + 1.21i)5-s + (−0.514 + 0.261i)6-s + (−0.129 + 0.194i)7-s + (−0.501 − 0.865i)8-s + (−0.277 + 0.185i)9-s + (−1.17 + 0.329i)10-s + (0.0887 − 0.0474i)11-s + (−0.346 − 0.461i)12-s + (−0.697 − 0.0686i)13-s + (−0.213 − 0.0939i)14-s + (−0.650 + 0.269i)15-s + (0.764 − 0.644i)16-s + (1.02 + 0.423i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0597486 - 1.23286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0597486 - 1.23286i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.246 - 1.39i)T \) |
| 3 | \( 1 + (-0.290 - 0.956i)T \) |
| good | 5 | \( 1 + (-0.267 - 2.71i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (0.343 - 0.513i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.294 + 0.157i)T + (6.11 - 9.14i)T^{2} \) |
| 13 | \( 1 + (2.51 + 0.247i)T + (12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-4.21 - 1.74i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (1.78 - 1.46i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-0.582 + 2.92i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.221 - 0.414i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (0.222 - 0.222i)T - 31iT^{2} \) |
| 37 | \( 1 + (7.22 - 8.79i)T + (-7.21 - 36.2i)T^{2} \) |
| 41 | \( 1 + (-3.19 - 0.636i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.347 + 1.14i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-2.91 + 7.03i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-4.77 - 8.93i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (0.308 - 0.0303i)T + (57.8 - 11.5i)T^{2} \) |
| 61 | \( 1 + (0.821 - 0.249i)T + (50.7 - 33.8i)T^{2} \) |
| 67 | \( 1 + (-12.6 + 3.82i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (-0.852 - 0.569i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-3.54 - 5.30i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.11 - 5.11i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (1.75 + 2.14i)T + (-16.1 + 81.4i)T^{2} \) |
| 89 | \( 1 + (2.44 + 12.3i)T + (-82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-5.61 + 5.61i)T - 97iT^{2} \) |
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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
−11.90651546960919111039580263164, −10.51579746004347501506676398072, −10.04031302639159555376706486807, −8.959025062965172032820612667703, −7.972910921980655899707178516082, −7.02612046666280880984033057401, −6.16280442142633735887434440334, −5.14850401443264645534404264815, −3.86328873086779321253952785746, −2.83284812549858147046385128821,
0.793452479991645454361236228569, 2.15418221173422278709884945245, 3.62632486201581595239392579012, 4.87242249421260268034067704535, 5.65957213174721280130154160778, 7.25031596407555916951734514180, 8.341350284894504687428688780694, 9.189103060221422996872901616155, 9.852446451802259616595603028388, 11.02095323100015239544698105285
