| L(s) = 1 | + (0.173 + 0.984i)2-s + (−1.60 − 0.651i)3-s + (−0.939 + 0.342i)4-s + (−0.122 + 0.336i)5-s + (0.362 − 1.69i)6-s + (−2.54 + 0.714i)7-s + (−0.5 − 0.866i)8-s + (2.15 + 2.09i)9-s + (−0.352 − 0.0621i)10-s − 4.13i·11-s + (1.73 + 0.0629i)12-s + (0.464 + 0.0819i)13-s + (−1.14 − 2.38i)14-s + (0.415 − 0.460i)15-s + (0.766 − 0.642i)16-s + (−1.75 − 2.08i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.926 − 0.375i)3-s + (−0.469 + 0.171i)4-s + (−0.0547 + 0.150i)5-s + (0.148 − 0.691i)6-s + (−0.962 + 0.270i)7-s + (−0.176 − 0.306i)8-s + (0.717 + 0.696i)9-s + (−0.111 − 0.0196i)10-s − 1.24i·11-s + (0.499 + 0.0181i)12-s + (0.128 + 0.0227i)13-s + (−0.306 − 0.637i)14-s + (0.107 − 0.118i)15-s + (0.191 − 0.160i)16-s + (−0.425 − 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.930683 + 0.151416i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930683 + 0.151416i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (1.60 + 0.651i)T \) |
| 7 | \( 1 + (2.54 - 0.714i)T \) |
| 19 | \( 1 + (-3.58 - 2.47i)T \) |
| good | 5 | \( 1 + (0.122 - 0.336i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + 4.13iT - 11T^{2} \) |
| 13 | \( 1 + (-0.464 - 0.0819i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.75 + 2.08i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.60 - 0.282i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.141 - 0.0513i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.771 - 0.445i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.04 + 4.64i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.78 - 10.1i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.95 + 3.32i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.29 + 8.68i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-11.6 + 4.22i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-3.59 + 3.01i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.59 - 9.03i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.59 - 0.282i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.88 - 2.41i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.136 + 0.0498i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (2.38 + 2.84i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.58 + 2.06i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.53 - 3.10i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.26 + 14.4i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.33234461660052854459593165796, −9.392929330024180105819258998362, −8.544694749670516214241467439179, −7.41349048450967983419918295635, −6.78896672433229880843688198162, −5.85604905383972588668248008318, −5.43215706572383266263613998799, −4.04818074827574814715659437780, −2.87407823978132477166264777256, −0.74402355667850430969336918424,
0.911688878679692301477919178924, 2.63919998768605086008446439576, 3.97688257917852080194079290296, 4.61743087679068817903706123394, 5.68674530529129965824687872109, 6.64614393733841104761848961448, 7.44564803882318205957495356331, 9.001756511320275893377982276311, 9.591023180659949782778035379518, 10.33859028162358185484562850909
