| L(s) = 1 | + (1.61 − 1.17i)2-s + (2.34 + 7.20i)3-s + (1.23 − 3.80i)4-s + (−4.37 − 3.17i)5-s + (12.2 + 8.90i)6-s + (−6.84 + 21.0i)7-s + (−2.47 − 7.60i)8-s + (−24.5 + 17.8i)9-s − 10.8·10-s + 30.3·12-s + (−62.1 + 45.1i)13-s + (13.6 + 42.1i)14-s + (12.6 − 38.9i)15-s + (−12.9 − 9.40i)16-s + (−47.9 − 34.8i)17-s + (−18.7 + 57.8i)18-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.450 + 1.38i)3-s + (0.154 − 0.475i)4-s + (−0.391 − 0.284i)5-s + (0.834 + 0.605i)6-s + (−0.369 + 1.13i)7-s + (−0.109 − 0.336i)8-s + (−0.910 + 0.661i)9-s − 0.342·10-s + 0.728·12-s + (−1.32 + 0.963i)13-s + (0.261 + 0.804i)14-s + (0.217 − 0.670i)15-s + (−0.202 − 0.146i)16-s + (−0.684 − 0.497i)17-s + (−0.245 + 0.756i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.742865 + 1.59249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.742865 + 1.59249i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 + 1.17i)T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + (-2.34 - 7.20i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (4.37 + 3.17i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (6.84 - 21.0i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (62.1 - 45.1i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (47.9 + 34.8i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-29.4 - 90.6i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (6.30 - 19.4i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (172. - 125. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (44.9 - 138. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-25.5 - 78.7i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-27.9 - 85.9i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-189. + 138. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-93.4 + 287. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-120. - 87.7i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 826.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-727. - 528. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (42.6 - 131. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-246. + 178. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-618. - 449. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-471. + 342. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−12.01312028668844011249299418985, −11.14046628608386033606039122796, −9.921316130368246290219653346678, −9.366610587103738939868716860895, −8.528857182409443104001405402954, −6.86817516675777193218120735420, −5.32049961123913331593678997115, −4.62791903579625435160938100921, −3.48862108786274895017614488872, −2.36535097834120101568169491666,
0.52555969208373491429745053604, 2.43708678727578445978575942477, 3.62456290783945569124048441430, 5.13175297021411804584580615832, 6.67824769186687546193411283450, 7.29360569856819832521750043613, 7.75456407076153818458744591321, 9.122919540441467402255182585533, 10.57305970638395099864920316670, 11.56973073578899657579519886821
