| L(s) = 1 | − 1.73i·3-s + 8.94i·5-s + (−1.52 + 6.83i)7-s − 2.99·9-s − 0.217·11-s + 17.2i·13-s + 15.4·15-s − 7.22i·17-s − 28.0i·19-s + (11.8 + 2.64i)21-s − 23.9·23-s − 54.9·25-s + 5.19i·27-s − 26.9·29-s − 27.0i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.78i·5-s + (−0.218 + 0.975i)7-s − 0.333·9-s − 0.0198·11-s + 1.32i·13-s + 1.03·15-s − 0.425i·17-s − 1.47i·19-s + (0.563 + 0.126i)21-s − 1.03·23-s − 2.19·25-s + 0.192i·27-s − 0.930·29-s − 0.873i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8034203596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8034203596\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (1.52 - 6.83i)T \) |
| good | 5 | \( 1 - 8.94iT - 25T^{2} \) |
| 11 | \( 1 + 0.217T + 121T^{2} \) |
| 13 | \( 1 - 17.2iT - 169T^{2} \) |
| 17 | \( 1 + 7.22iT - 289T^{2} \) |
| 19 | \( 1 + 28.0iT - 361T^{2} \) |
| 23 | \( 1 + 23.9T + 529T^{2} \) |
| 29 | \( 1 + 26.9T + 841T^{2} \) |
| 31 | \( 1 + 27.0iT - 961T^{2} \) |
| 37 | \( 1 + 17.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 56.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 75.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.03T + 2.80e3T^{2} \) |
| 59 | \( 1 - 2.32iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 60.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 102.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 22.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 107. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 141.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 81.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 167. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 53.8iT - 9.40e3T^{2} \) |
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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
−11.04533452673681425646393232309, −9.711571007504257821057370307147, −9.140756938550380324310333115364, −7.88054499895042514509105515684, −6.99182310665546848755313368316, −6.48830925499597247127465730723, −5.59436902741474274247267933904, −4.01491376246071734007915314980, −2.72664356445420621342010570178, −2.16332848411649135579160362239,
0.28016276691412789450903501518, 1.53835670466404456491870172441, 3.56554743633055060611564597829, 4.25043898025678070152884047326, 5.30197805969504037817501509955, 5.94742659487184637889221946497, 7.59387311486476284678859373188, 8.228952731832327384706822678059, 9.006886160434280087462327185535, 10.09751451244131935173685149137
