| L(s) = 1 | − i·2-s − 4-s − 2.82i·7-s + i·8-s − 5.65·11-s + i·13-s − 2.82·14-s + 16-s − 0.828i·17-s − 2.82·19-s + 5.65i·22-s − 8.48i·23-s + 26-s + 2.82i·28-s − 8.82·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.06i·7-s + 0.353i·8-s − 1.70·11-s + 0.277i·13-s − 0.755·14-s + 0.250·16-s − 0.200i·17-s − 0.648·19-s + 1.20i·22-s − 1.76i·23-s + 0.196·26-s + 0.534i·28-s − 1.63·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1394589198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1394589198\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 11.6iT - 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 9.65iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 14.4iT - 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 6.34iT - 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 3.17iT - 97T^{2} \) |
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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
−8.243744818060668311793564356144, −7.53245803558132507987160795811, −7.03561497551526946568799356623, −5.92625595043370380252814307354, −5.27281568658009439963643129504, −4.26895663859179361764603997821, −4.00611540142545331967831319057, −2.68963572220851428393340274902, −2.27940033874780343911538634731, −0.866926159922866279614516790949,
0.04300166957174082848059122014, 1.74003027819455224080666334692, 2.69979256197105389840634587459, 3.46586058128657552128552129534, 4.59469192051395935721109801310, 5.39378798042366188740281144602, 5.64448036419382053088899486559, 6.49914714070979670350513812987, 7.37054154193364408412485294135, 8.160721808417514821924584788924
