L(s) = 1 | − i·2-s − 4-s + 0.929·5-s + (−2.07 + 1.63i)7-s + i·8-s − 0.929i·10-s + i·11-s + (1.63 + 2.07i)14-s + 16-s − 2.34·17-s − 6.87i·19-s − 0.929·20-s + 22-s + 3.04i·23-s − 4.13·25-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.415·5-s + (−0.785 + 0.619i)7-s + 0.353i·8-s − 0.294i·10-s + 0.301i·11-s + (0.437 + 0.555i)14-s + 0.250·16-s − 0.569·17-s − 1.57i·19-s − 0.207·20-s + 0.213·22-s + 0.635i·23-s − 0.827·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0523 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0523 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6149948469\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6149948469\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.07 - 1.63i)T \) |
| 11 | \( 1 - iT \) |
good | 5 | \( 1 - 0.929T + 5T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 + 6.87iT - 19T^{2} \) |
| 23 | \( 1 - 3.04iT - 23T^{2} \) |
| 29 | \( 1 - 6.39iT - 29T^{2} \) |
| 31 | \( 1 - 10.0iT - 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.15T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 + 0.0878T + 67T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 4.06T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 15.6iT - 97T^{2} \) |
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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
−9.790202759411511018132792599972, −9.040050939115512683272694284726, −8.621507490930312411826907581309, −7.20594376333783704073141459509, −6.55061747485704364528393954632, −5.45388163549011253809594005566, −4.76350709135407470611765993586, −3.47118956810352184933443199300, −2.69670092259974169606776059556, −1.60222225534461171303534266997,
0.23962020225016066391790956021, 2.00412784936308371621538383241, 3.49766467689644574256715391732, 4.19992945404993743800534825904, 5.40687299885042985088687147085, 6.23932144138914177020585190134, 6.69498838651167334543651160987, 7.85320340598134582747319891128, 8.310216966442115966974285251321, 9.616692430225485365789504549196