| L(s) = 1 | + (1.06 + 0.775i)2-s + (2.12 − 6.54i)3-s + (−1.93 − 5.95i)4-s + (−8.06 + 5.85i)5-s + (7.34 − 5.33i)6-s + (−2.16 − 6.65i)7-s + (5.81 − 17.8i)8-s + (−16.4 − 11.9i)9-s − 13.1·10-s + (−29.4 − 21.5i)11-s − 43.0·12-s + (24.2 + 17.6i)13-s + (2.85 − 8.78i)14-s + (21.1 + 65.1i)15-s + (−20.4 + 14.8i)16-s + (73.5 − 53.4i)17-s + ⋯ |
| L(s) = 1 | + (0.377 + 0.274i)2-s + (0.409 − 1.25i)3-s + (−0.241 − 0.744i)4-s + (−0.720 + 0.523i)5-s + (0.499 − 0.362i)6-s + (−0.116 − 0.359i)7-s + (0.256 − 0.790i)8-s + (−0.609 − 0.442i)9-s − 0.415·10-s + (−0.806 − 0.590i)11-s − 1.03·12-s + (0.517 + 0.375i)13-s + (0.0544 − 0.167i)14-s + (0.364 + 1.12i)15-s + (−0.319 + 0.232i)16-s + (1.04 − 0.762i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.947959 - 1.27723i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.947959 - 1.27723i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.16 + 6.65i)T \) |
| 11 | \( 1 + (29.4 + 21.5i)T \) |
| good | 2 | \( 1 + (-1.06 - 0.775i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (-2.12 + 6.54i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (8.06 - 5.85i)T + (38.6 - 118. i)T^{2} \) |
| 13 | \( 1 + (-24.2 - 17.6i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-73.5 + 53.4i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-21.0 + 64.6i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-73.6 - 226. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (134. + 97.7i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-72.9 - 224. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-39.5 + 121. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-148. + 456. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-201. - 146. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-101. - 311. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (163. - 118. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 - 424.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (82.6 - 60.0i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-51.0 - 157. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (694. + 504. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-418. + 304. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 343.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (43.8 + 31.8i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−13.57056385904838145969746256691, −13.05578865927495126161143079214, −11.55372828855602673756431881608, −10.48251428915929385850834471592, −8.862192390411980773730379885050, −7.42236870644049784593222791447, −6.81840423305919857028275258829, −5.20980169960617315626002853466, −3.19679386036137531205752233014, −0.940650183340402108290226750063,
3.13284480677701065169112807429, 4.15438261000686898044335628956, 5.24997041540443165144505844984, 7.79864628661482520081390263263, 8.603586205780953441053805115584, 9.823623406252226243532351086485, 11.01458606854136982783073663978, 12.31485362162825554173842433944, 12.97422555455259350414297822979, 14.46461286012977960537833577079
