L(s) = 1 | + (2.5 + 4.33i)5-s + (−16.8 + 29.1i)7-s + (−4.74 + 8.22i)11-s + (−19.8 − 34.3i)13-s − 106.·17-s − 96.5·19-s + (20.0 + 34.6i)23-s + (−12.5 + 21.6i)25-s + (126. − 218. i)29-s + (−24.4 − 42.3i)31-s − 168.·35-s − 136.·37-s + (69.5 + 120. i)41-s + (101. − 175. i)43-s + (150. − 259. i)47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (−0.909 + 1.57i)7-s + (−0.130 + 0.225i)11-s + (−0.423 − 0.733i)13-s − 1.51·17-s − 1.16·19-s + (0.181 + 0.314i)23-s + (−0.100 + 0.173i)25-s + (0.808 − 1.39i)29-s + (−0.141 − 0.245i)31-s − 0.813·35-s − 0.606·37-s + (0.264 + 0.458i)41-s + (0.358 − 0.621i)43-s + (0.465 − 0.806i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8207275388\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8207275388\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 7 | \( 1 + (16.8 - 29.1i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (4.74 - 8.22i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (19.8 + 34.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 96.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-20.0 - 34.6i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-126. + 218. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (24.4 + 42.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 136.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-69.5 - 120. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-101. + 175. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-150. + 259. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 344.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (30.4 + 52.7i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-61.0 + 105. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-372. - 644. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 436.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 586.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (236. - 410. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (499. - 865. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 204.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-816. + 1.41e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−8.859288581579308516073015637591, −8.418268792987705681976672942805, −7.19738929703902738247619326464, −6.39194346716146437874223468898, −5.84448276001467470936685290865, −4.92133486709111223845216680978, −3.77837751325313850283036367636, −2.45422041817290260652168076528, −2.35284044220657609937701660252, −0.24518963504275157800621430020,
0.70200045830276321285471631959, 1.97487690371201575242372545517, 3.18058234027122998657035786221, 4.26692996345239900189696626756, 4.66804922008152497248308555005, 6.09104227155947717807156428416, 6.81855819700399602751702554629, 7.24716605505514385232161816488, 8.512297161857726672028463589528, 9.071136415352080737339081034030