| L(s) = 1 | + (1.99 − 1.44i)2-s + (−0.165 − 0.509i)3-s + (−0.595 + 1.83i)4-s + (1.24 + 0.902i)5-s + (−1.06 − 0.776i)6-s + (8.72 − 26.8i)7-s + (7.55 + 23.2i)8-s + (21.6 − 15.7i)9-s + 3.78·10-s + 1.03·12-s + (55.3 − 40.2i)13-s + (−21.5 − 66.2i)14-s + (0.254 − 0.782i)15-s + (36.2 + 26.3i)16-s + (44.7 + 32.5i)17-s + (20.3 − 62.6i)18-s + ⋯ |
| L(s) = 1 | + (0.704 − 0.512i)2-s + (−0.0318 − 0.0980i)3-s + (−0.0744 + 0.229i)4-s + (0.111 + 0.0807i)5-s + (−0.0726 − 0.0528i)6-s + (0.471 − 1.45i)7-s + (0.334 + 1.02i)8-s + (0.800 − 0.581i)9-s + 0.119·10-s + 0.0248·12-s + (1.18 − 0.858i)13-s + (−0.410 − 1.26i)14-s + (0.00437 − 0.0134i)15-s + (0.567 + 0.411i)16-s + (0.638 + 0.464i)17-s + (0.266 − 0.819i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.25817 - 1.05338i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25817 - 1.05338i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.99 + 1.44i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (0.165 + 0.509i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-1.24 - 0.902i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-8.72 + 26.8i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-55.3 + 40.2i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-44.7 - 32.5i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-17.0 - 52.4i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (34.9 - 107. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (57.2 - 41.6i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (65.0 - 200. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (59.3 + 182. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-158. - 487. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-303. + 220. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (156. - 481. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (379. + 275. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 289.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-318. - 231. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (89.4 - 275. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (137. - 99.6i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (245. + 178. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (519. - 377. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−12.83249955516554639246920198101, −12.04729284463629418244113502778, −10.77409481353454896457950660329, −10.14051294610166054887057954818, −8.298258323613226458769963973218, −7.47138545906802460004730596288, −5.91452375616785673136223808152, −4.24372142393682277717942595798, −3.56611539802340981410022445091, −1.36084750261072559631667349327,
1.81438664995222520487498439702, 4.05679747545047164090875624994, 5.29246225320323497919830458500, 6.07416977133056303322254687274, 7.49728259329113430708089242821, 8.914071238144201703754724518436, 9.859080026950262789854416554559, 11.22308148643658212835619432743, 12.24604212998938890851275573963, 13.40896217391517652701611747548
