| L(s) = 1 | + (0.903 + 2.78i)2-s + (−3.67 − 2.66i)3-s + (−0.441 + 0.320i)4-s + (1.90 − 5.87i)5-s + (4.10 − 12.6i)6-s + (−25.9 + 18.8i)7-s + (17.6 + 12.8i)8-s + (−1.97 − 6.06i)9-s + 18.0·10-s + (35.3 + 8.87i)11-s + 2.47·12-s + (−1.38 − 4.27i)13-s + (−75.9 − 55.2i)14-s + (−22.6 + 16.4i)15-s + (−21.0 + 64.7i)16-s + (18.3 − 56.5i)17-s + ⋯ |
| L(s) = 1 | + (0.319 + 0.982i)2-s + (−0.707 − 0.513i)3-s + (−0.0552 + 0.0401i)4-s + (0.170 − 0.525i)5-s + (0.279 − 0.859i)6-s + (−1.40 + 1.01i)7-s + (0.779 + 0.566i)8-s + (−0.0729 − 0.224i)9-s + 0.571·10-s + (0.969 + 0.243i)11-s + 0.0596·12-s + (−0.0296 − 0.0912i)13-s + (−1.45 − 1.05i)14-s + (−0.390 + 0.283i)15-s + (−0.328 + 1.01i)16-s + (0.262 − 0.807i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.850245 + 0.286943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.850245 + 0.286943i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-35.3 - 8.87i)T \) |
| good | 2 | \( 1 + (-0.903 - 2.78i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (3.67 + 2.66i)T + (8.34 + 25.6i)T^{2} \) |
| 5 | \( 1 + (-1.90 + 5.87i)T + (-101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (25.9 - 18.8i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (1.38 + 4.27i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-18.3 + 56.5i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (23.4 + 17.0i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 38.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (31.9 - 23.1i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (82.2 + 253. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (90.7 - 65.9i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-108. - 78.8i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-147. - 107. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-13.2 - 40.6i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-146. + 106. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (172. - 532. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 770.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (8.20 - 25.2i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (301. - 218. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (78.1 + 240. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-326. + 1.00e3i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 - 58.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + (184. + 568. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−20.18626442735732087901253267720, −18.75424984197414130165426614172, −17.14922475520902071137488495591, −16.21594354894836489091281577417, −14.91503236707408509257860359613, −13.05073072972537014866518991248, −11.81370896096322080741373374398, −9.272180715264379182101838193518, −6.77966990996318756938360369677, −5.70815067538849241204358239022,
3.73900516887135342538893335489, 6.66160716258020432602206536037, 10.07955011400896467346673578350, 10.89021947794851228552809811572, 12.47149493153801801648877272451, 13.92755997292439631322226430791, 16.20731881589014501790331676692, 16.98166003008091623413204992310, 19.20498759816674962523536112977, 19.98300628792310219331538333646
