| L(s) = 1 | + (−0.579 − 0.210i)2-s + (−5.83 − 4.89i)4-s + (−9.88 + 8.29i)5-s + (6.05 − 10.4i)7-s + (4.81 + 8.33i)8-s + (7.46 − 2.71i)10-s + (−1.51 − 2.61i)11-s + (−2.02 + 11.4i)13-s + (−5.71 + 4.79i)14-s + (9.55 + 54.1i)16-s + (104. + 38.0i)17-s + (82.6 + 5.09i)19-s + 98.2·20-s + (0.323 + 1.83i)22-s + (0.581 + 0.487i)23-s + ⋯ |
| L(s) = 1 | + (−0.204 − 0.0745i)2-s + (−0.729 − 0.612i)4-s + (−0.883 + 0.741i)5-s + (0.327 − 0.566i)7-s + (0.212 + 0.368i)8-s + (0.236 − 0.0859i)10-s + (−0.0414 − 0.0717i)11-s + (−0.0432 + 0.245i)13-s + (−0.109 + 0.0916i)14-s + (0.149 + 0.846i)16-s + (1.49 + 0.542i)17-s + (0.998 + 0.0615i)19-s + 1.09·20-s + (0.00313 + 0.0177i)22-s + (0.00527 + 0.00442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.03516 + 0.222296i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03516 + 0.222296i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (-82.6 - 5.09i)T \) |
| good | 2 | \( 1 + (0.579 + 0.210i)T + (6.12 + 5.14i)T^{2} \) |
| 5 | \( 1 + (9.88 - 8.29i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-6.05 + 10.4i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (1.51 + 2.61i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (2.02 - 11.4i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-104. - 38.0i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-0.581 - 0.487i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-96.6 + 35.1i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (59.9 - 103. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 336.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-17.2 - 97.7i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-88.1 + 73.9i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (298. - 108. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (-244. - 205. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (225. + 82.0i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (166. + 139. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (510. - 185. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (166. - 139. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (99.2 + 562. i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-185. - 1.05e3i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-315. + 546. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-23.4 + 132. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-859. - 312. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−12.21899247749999521552451189420, −11.19288352854761120962409998142, −10.39922162254199372198733780801, −9.491960714891040069810899133018, −8.120259911349093693171960938823, −7.39316973861251303655313370229, −5.89297178210230421528191199517, −4.55305625485562140614899612708, −3.37282018603240960702565629712, −1.07943074402381090766317057774,
0.72115796093286082951884384605, 3.20848312704402519557372417701, 4.49524509510710501345649676819, 5.47959516469762823080652283354, 7.49937694031430002672337696516, 8.076476577900718150791436118350, 9.032632708866025095482398097163, 9.955882441929947563567046811393, 11.64606948301759569179272964456, 12.14725519963162369746560156329
