| L(s) = 1 | − 4.34·5-s + (−0.646 − 6.97i)7-s − 7.76·11-s − 13.4i·13-s + 28.9·17-s + 32.7·19-s − 16.8·23-s − 6.12·25-s + 38.3i·29-s − 50.1·31-s + (2.80 + 30.2i)35-s − 39.1·37-s − 63.0·41-s − 7.01i·43-s − 23.4i·47-s + ⋯ |
| L(s) = 1 | − 0.868·5-s + (−0.0922 − 0.995i)7-s − 0.705·11-s − 1.03i·13-s + 1.70·17-s + 1.72·19-s − 0.732·23-s − 0.245·25-s + 1.32i·29-s − 1.61·31-s + (0.0801 + 0.865i)35-s − 1.05·37-s − 1.53·41-s − 0.163i·43-s − 0.499i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.332i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.942 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1667733855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1667733855\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.646 + 6.97i)T \) |
| good | 5 | \( 1 + 4.34T + 25T^{2} \) |
| 11 | \( 1 + 7.76T + 121T^{2} \) |
| 13 | \( 1 + 13.4iT - 169T^{2} \) |
| 17 | \( 1 - 28.9T + 289T^{2} \) |
| 19 | \( 1 - 32.7T + 361T^{2} \) |
| 23 | \( 1 + 16.8T + 529T^{2} \) |
| 29 | \( 1 - 38.3iT - 841T^{2} \) |
| 31 | \( 1 + 50.1T + 961T^{2} \) |
| 37 | \( 1 + 39.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 63.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 7.01iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 23.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 90.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 78.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 49.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 15.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 62.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 59.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 137. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 26.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 105. iT - 9.40e3T^{2} \) |
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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
−9.436047804990329077497994675330, −8.167305386717536007153295241235, −7.60391935691678231021314895817, −7.16943318961203096442703193234, −5.63266993292253769475315436805, −5.04842742800956539730158965324, −3.54849012644120944459495167766, −3.31964504195268851983291464035, −1.30145098996130604475378233875, −0.05540276505433655388862352708,
1.73712076953272454785031983933, 3.07504412074344571683928796995, 3.88388113005839716031958006626, 5.21480876002073776532185074907, 5.73647406530740263526715138316, 7.04586866117343731246294076844, 7.79561297318968020342318864941, 8.425968786063050485315017390886, 9.527615135555475108223949824744, 9.993640209966313790137991754207
