L(s) = 1 | + (−1 + 1.73i)2-s + (−4.69 − 8.12i)3-s + (−1.99 − 3.46i)4-s + (4.69 − 8.12i)5-s + 18.7·6-s + 7.99·8-s + (−30.5 + 52.8i)9-s + (9.38 + 16.2i)10-s + (−10 − 17.3i)11-s + (−18.7 + 32.4i)12-s − 65.6·13-s − 88.0·15-s + (−8 + 13.8i)16-s + (28.1 + 48.7i)17-s + (−61 − 105. i)18-s + (4.69 − 8.12i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.902 − 1.56i)3-s + (−0.249 − 0.433i)4-s + (0.419 − 0.726i)5-s + 1.27·6-s + 0.353·8-s + (−1.12 + 1.95i)9-s + (0.296 + 0.513i)10-s + (−0.274 − 0.474i)11-s + (−0.451 + 0.781i)12-s − 1.40·13-s − 1.51·15-s + (−0.125 + 0.216i)16-s + (0.401 + 0.695i)17-s + (−0.798 − 1.38i)18-s + (0.0566 − 0.0980i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0229422 + 0.361522i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0229422 + 0.361522i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (4.69 + 8.12i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (-4.69 + 8.12i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (10 + 17.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-28.1 - 48.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-4.69 + 8.12i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (24 - 41.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + (103. + 178. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-39 + 67.5i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-103. + 178. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (31 + 53.6i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (333. + 576. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-136. + 235. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (290 + 502. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 544T + 3.57e5T^{2} \) |
| 73 | \( 1 + (300. + 519. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-340 + 588. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (750. - 1.29e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 656.T + 9.12e5T^{2} \) |
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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
−12.89713230748718309861987177058, −12.08708230803099149625628226931, −10.87130390184026256107190888922, −9.409183659009238899853363886952, −8.008549585650892827207378958293, −7.23496980258033751371402825953, −5.94545674548936142305268069061, −5.20012393899947765260331933286, −1.81350884353071380397138412146, −0.25101696356901132321400637820,
2.85843559049178382146361478411, 4.43314697378421512934346563288, 5.53957525570828792871722273548, 7.18310907878607353881183457087, 9.146562065883587131643315769404, 10.04176033724342837725100015557, 10.48890439157576424321597970986, 11.57906667667770828787321147779, 12.45752750190181179968033367445, 14.25355213940592940117272383602