| L(s) = 1 | + (1.86 + 0.714i)2-s + (1.35 + 1.07i)3-s + (2.97 + 2.66i)4-s + (−2.69 − 1.29i)5-s + (1.75 + 2.98i)6-s + (6.11 + 4.87i)7-s + (3.65 + 7.11i)8-s + (0.667 + 2.92i)9-s + (−4.10 − 4.34i)10-s + (7.32 + 1.67i)11-s + (1.15 + 6.83i)12-s + (2.97 − 13.0i)13-s + (7.93 + 13.4i)14-s + (−2.24 − 4.66i)15-s + (1.75 + 15.9i)16-s − 23.3·17-s + ⋯ |
| L(s) = 1 | + (0.934 + 0.357i)2-s + (0.451 + 0.359i)3-s + (0.744 + 0.667i)4-s + (−0.538 − 0.259i)5-s + (0.293 + 0.497i)6-s + (0.873 + 0.696i)7-s + (0.457 + 0.889i)8-s + (0.0741 + 0.324i)9-s + (−0.410 − 0.434i)10-s + (0.666 + 0.152i)11-s + (0.0960 + 0.569i)12-s + (0.228 − 1.00i)13-s + (0.566 + 0.962i)14-s + (−0.149 − 0.310i)15-s + (0.109 + 0.993i)16-s − 1.37·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.71585 + 2.06269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.71585 + 2.06269i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.86 - 0.714i)T \) |
| 3 | \( 1 + (-1.35 - 1.07i)T \) |
| 29 | \( 1 + (-16.0 - 24.1i)T \) |
| good | 5 | \( 1 + (2.69 + 1.29i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (-6.11 - 4.87i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (-7.32 - 1.67i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (-2.97 + 13.0i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + 23.3T + 289T^{2} \) |
| 19 | \( 1 + (6.04 - 4.82i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (-17.5 - 36.3i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-20.2 + 42.1i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (5.42 + 23.7i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 - 44.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (0.174 + 0.362i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (41.7 + 9.51i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (82.2 + 39.6i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 111. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (-24.6 + 30.8i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (-55.8 + 12.7i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (0.0372 + 0.00851i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-80.1 + 38.5i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (94.4 - 21.5i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (-81.6 + 65.1i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-16.2 - 7.81i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (0.204 + 0.256i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.45375733881359137099684441389, −11.02088533076254112604590610627, −9.433499034916618153103447835414, −8.351500668615404594014355976111, −7.85615471414170783729035910799, −6.53802016778302691079595580295, −5.33823812215473947432617685297, −4.48475070754166789958189959257, −3.45546485800496714870187418247, −2.05483399756251510038546316351,
1.29940217055333644982005692865, 2.66736906695537624587480197274, 4.13005341843332745637270335240, 4.60326548285309052905020707227, 6.49156749295411875540030625967, 6.93726407032258718857312703781, 8.176134569240456027814350955429, 9.196499409917633125956021101885, 10.62582662004968169216971971741, 11.23854936009106312781678134245
