L(s) = 1 | + (−1.63 + 0.577i)3-s + (−2.15 − 0.603i)5-s + 1.63·7-s + (2.33 − 1.88i)9-s − 0.207·11-s − 1.70i·13-s + (3.86 − 0.256i)15-s + 5.57i·17-s − 2.31i·19-s + (−2.67 + 0.945i)21-s + (−2.87 + 3.83i)23-s + (4.27 + 2.60i)25-s + (−2.72 + 4.42i)27-s − 4.70i·29-s − 2.76·31-s + ⋯ |
L(s) = 1 | + (−0.942 + 0.333i)3-s + (−0.962 − 0.270i)5-s + 0.618·7-s + (0.777 − 0.628i)9-s − 0.0624·11-s − 0.472i·13-s + (0.997 − 0.0663i)15-s + 1.35i·17-s − 0.532i·19-s + (−0.583 + 0.206i)21-s + (−0.600 + 0.799i)23-s + (0.854 + 0.520i)25-s + (−0.523 + 0.851i)27-s − 0.873i·29-s − 0.496·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.545 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4101294135\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4101294135\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.63 - 0.577i)T \) |
| 5 | \( 1 + (2.15 + 0.603i)T \) |
| 23 | \( 1 + (2.87 - 3.83i)T \) |
good | 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 0.207T + 11T^{2} \) |
| 13 | \( 1 + 1.70iT - 13T^{2} \) |
| 17 | \( 1 - 5.57iT - 17T^{2} \) |
| 19 | \( 1 + 2.31iT - 19T^{2} \) |
| 29 | \( 1 + 4.70iT - 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 + 7.81iT - 41T^{2} \) |
| 43 | \( 1 + 3.48T + 43T^{2} \) |
| 47 | \( 1 - 6.72T + 47T^{2} \) |
| 53 | \( 1 + 4.72iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 8.69iT - 61T^{2} \) |
| 67 | \( 1 + 9.87T + 67T^{2} \) |
| 71 | \( 1 - 0.717iT - 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 4.32iT - 79T^{2} \) |
| 83 | \( 1 - 1.66iT - 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 9.51T + 97T^{2} \) |
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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
−9.358930786826274771032564303279, −8.364782481738718460917559748416, −7.75010004966906852932637140199, −6.85043500748844969737090996240, −5.83196657440269562685872642066, −5.09233851277530695464763522521, −4.21174742592398237544745191506, −3.51289983824839386515510360429, −1.67316045459781342416066656154, −0.20983374543532406540198213137,
1.28667892960691057580520255922, 2.72775396974527720772861862103, 4.12453565293299828646902286682, 4.76805501016077609312051221461, 5.68943177166239849108906897786, 6.75213253457854833471812347669, 7.32596550936358642082166142814, 8.042571006944895149724728292510, 8.951797440923239627765579178520, 10.09868052757726126844558258148