| L(s) = 1 | + (−1.61 + 1.17i)2-s + (2.75 + 8.48i)3-s + (1.23 − 3.80i)4-s + (−4.04 − 2.93i)5-s + (−14.4 − 10.4i)6-s + (−4.06 + 12.5i)7-s + (2.47 + 7.60i)8-s + (−42.4 + 30.8i)9-s + 10·10-s + (−22.6 + 28.6i)11-s + 35.6·12-s + (30.1 − 21.9i)13-s + (−8.13 − 25.0i)14-s + (13.7 − 42.4i)15-s + (−12.9 − 9.40i)16-s + (−89.8 − 65.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.530 + 1.63i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (−0.981 − 0.713i)6-s + (−0.219 + 0.676i)7-s + (0.109 + 0.336i)8-s + (−1.57 + 1.14i)9-s + 0.316·10-s + (−0.620 + 0.784i)11-s + 0.858·12-s + (0.643 − 0.467i)13-s + (−0.155 − 0.478i)14-s + (0.237 − 0.729i)15-s + (−0.202 − 0.146i)16-s + (−1.28 − 0.931i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.111396 - 0.889443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.111396 - 0.889443i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 - 1.17i)T \) |
| 5 | \( 1 + (4.04 + 2.93i)T \) |
| 11 | \( 1 + (22.6 - 28.6i)T \) |
| good | 3 | \( 1 + (-2.75 - 8.48i)T + (-21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + (4.06 - 12.5i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-30.1 + 21.9i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (89.8 + 65.3i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-4.51 - 13.9i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (27.4 - 84.3i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (187. - 136. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (35.0 - 107. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (76.4 + 235. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 515.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-140. - 433. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (423. - 307. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (205. - 633. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-534. - 388. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 250.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (480. + 348. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (114. - 351. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (42.8 - 31.1i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-97.6 - 70.9i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 308.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.13e3 + 827. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−14.14858672599048319745728969917, −12.73754488324799772949984363602, −11.14943607240331795881868248875, −10.41793934802532273933545290692, −9.102542136718754068826686876244, −8.895655914603639466237322029325, −7.39608625150754877526191142022, −5.51112973487993451903397347953, −4.48569264321580585308730226090, −2.83698525724171550229723781892,
0.53945698517672758868091614437, 2.14656679710626155014866764338, 3.61592363421056632079031212342, 6.31550153523085924872067962589, 7.23303017381603742202126246468, 8.166510661083178972841997063705, 9.031537116189002221348299621544, 10.81391341813184069843814777204, 11.45956454155948261106852707268, 12.96853072122802934594090747551
