L(s) = 1 | + 5.09·2-s + 17.9·4-s + (3.42 + 5.92i)5-s + (−0.241 + 18.5i)7-s + 50.6·8-s + (17.4 + 30.1i)10-s + (−20.5 + 35.5i)11-s + (31.8 − 55.2i)13-s + (−1.23 + 94.3i)14-s + 114.·16-s + (−38.0 − 65.8i)17-s + (46.7 − 81.0i)19-s + (61.3 + 106. i)20-s + (−104. + 181. i)22-s + (21.0 + 36.4i)23-s + ⋯ |
L(s) = 1 | + 1.80·2-s + 2.24·4-s + (0.305 + 0.529i)5-s + (−0.0130 + 0.999i)7-s + 2.24·8-s + (0.550 + 0.954i)10-s + (−0.562 + 0.974i)11-s + (0.680 − 1.17i)13-s + (−0.0234 + 1.80i)14-s + 1.79·16-s + (−0.542 − 0.939i)17-s + (0.564 − 0.978i)19-s + (0.686 + 1.18i)20-s + (−1.01 + 1.75i)22-s + (0.190 + 0.330i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.88692 + 1.24835i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.88692 + 1.24835i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.241 - 18.5i)T \) |
good | 2 | \( 1 - 5.09T + 8T^{2} \) |
| 5 | \( 1 + (-3.42 - 5.92i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (20.5 - 35.5i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-31.8 + 55.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (38.0 + 65.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-46.7 + 81.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-21.0 - 36.4i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (21.9 + 37.9i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 200.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (52.1 - 90.3i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-108. + 188. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-236. - 409. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 17.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + (211. + 365. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 154.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 838.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 940.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-65.8 - 113. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 620.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (102. + 177. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (432. - 749. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-331. - 573. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.52282040108092282006926532795, −11.46088270966164813424381110210, −10.70383670918535648950508938229, −9.335207560312655380622661146569, −7.64247459566386838267727136322, −6.59192894069885667126404245223, −5.55968724840907553698978961547, −4.79268285813109447715877332517, −3.13651030359737801051400587292, −2.34143396445639404175474095653,
1.62613967323751785415608664552, 3.45236768278967973249431626839, 4.28983844841427263367113609425, 5.51267386225471606336675450968, 6.37890263511291994288981795474, 7.53641973531072030850228516746, 8.971501573674485677886350886744, 10.66696237190502790747993997042, 11.16700003328824526477828672377, 12.41596316496579446139113188664