L(s) = 1 | − 3.02i·3-s − 1.69i·5-s + 7-s − 6.12·9-s + 1.32i·11-s + 1.69i·13-s − 5.12·15-s + 2·17-s − 3.02i·19-s − 3.02i·21-s − 5.12·23-s + 2.12·25-s + 9.43i·27-s − 6.04i·29-s + 10.2·31-s + ⋯ |
L(s) = 1 | − 1.74i·3-s − 0.758i·5-s + 0.377·7-s − 2.04·9-s + 0.399i·11-s + 0.470i·13-s − 1.32·15-s + 0.485·17-s − 0.692i·19-s − 0.659i·21-s − 1.06·23-s + 0.424·25-s + 1.81i·27-s − 1.12i·29-s + 1.84·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.562052 - 1.04626i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.562052 - 1.04626i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 3.02iT - 3T^{2} \) |
| 5 | \( 1 + 1.69iT - 5T^{2} \) |
| 11 | \( 1 - 1.32iT - 11T^{2} \) |
| 13 | \( 1 - 1.69iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 3.02iT - 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 + 6.04iT - 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 6.04iT - 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 1.32iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.64iT - 53T^{2} \) |
| 59 | \( 1 - 0.371iT - 59T^{2} \) |
| 61 | \( 1 - 1.69iT - 61T^{2} \) |
| 67 | \( 1 - 11.5iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 5.66iT - 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−12.02760016209081869353038293576, −11.49159043456510829847509753241, −9.877488026660716109241297158765, −8.570551349838598725734842522995, −7.943827082700113966125576336216, −6.91466619904657404457790069855, −5.94554954478163185201880575076, −4.59296494557860739664957514443, −2.44699884858925101646363558542, −1.10029157220730630631749103386,
2.94534133514974689881576895124, 3.95544329584768073844385342451, 5.14750988930028947081318611008, 6.17370794285566875514273294492, 7.83481488499514051033969834975, 8.835398432905057936421975266384, 9.968688227871516142777820088516, 10.51698832503969239090846631852, 11.24995763869216456485301347960, 12.32239993799148269669245319068