| L(s) = 1 | + i·2-s − 4-s + (1.48 + 1.67i)5-s + 3.35i·7-s − i·8-s + (−1.67 + 1.48i)10-s + 4·11-s − 0.387i·13-s − 3.35·14-s + 16-s + i·17-s + 5.92·19-s + (−1.48 − 1.67i)20-s + 4i·22-s + 1.35i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.662 + 0.749i)5-s + 1.26i·7-s − 0.353i·8-s + (−0.529 + 0.468i)10-s + 1.20·11-s − 0.107i·13-s − 0.895·14-s + 0.250·16-s + 0.242i·17-s + 1.35·19-s + (−0.331 − 0.374i)20-s + 0.852i·22-s + 0.281i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.937742821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.937742821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.48 - 1.67i)T \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 - 3.35iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 0.387iT - 13T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 - 1.35iT - 23T^{2} \) |
| 29 | \( 1 + 2.96T + 29T^{2} \) |
| 31 | \( 1 - 5.35T + 31T^{2} \) |
| 37 | \( 1 + 1.03iT - 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 + 4.38iT - 43T^{2} \) |
| 47 | \( 1 - 6.70iT - 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 7.61iT - 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.92iT - 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 17.5iT - 97T^{2} \) |
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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
−9.527399603757999687257326440654, −9.040347504292444863460320207989, −8.165400000184640037247434709725, −7.15877991994986797995821331516, −6.46997230474933666714812495463, −5.76231668060962214573138593921, −5.13729695385954344441267656852, −3.76422359545536209378676651322, −2.81513284846855175162324334167, −1.56881760973680539037186930177,
0.860597607670620124714320757569, 1.60463336513277830042178161428, 3.08776856048981778753720868911, 4.09889530082656828826569121463, 4.73702336966055990540779957230, 5.77482879594730413168279381917, 6.76114292991961032536968238200, 7.60548849602754104534632598415, 8.641473318806026633229498740686, 9.330110916768530596754491148649
