L(s) = 1 | + (5.18 − 2.99i)2-s + (−1.73 + 2.99i)3-s + (9.90 − 17.1i)4-s + (15.6 − 19.5i)5-s + 20.7i·6-s + (18.5 − 45.3i)7-s − 22.8i·8-s + (34.5 + 59.7i)9-s + (22.6 − 147. i)10-s + (−90.7 + 157. i)11-s + (34.2 + 59.3i)12-s − 258.·13-s + (−39.6 − 290. i)14-s + (31.4 + 80.6i)15-s + (90.1 + 156. i)16-s + (−21.6 + 37.4i)17-s + ⋯ |
L(s) = 1 | + (1.29 − 0.748i)2-s + (−0.192 + 0.333i)3-s + (0.619 − 1.07i)4-s + (0.625 − 0.780i)5-s + 0.575i·6-s + (0.378 − 0.925i)7-s − 0.356i·8-s + (0.426 + 0.737i)9-s + (0.226 − 1.47i)10-s + (−0.749 + 1.29i)11-s + (0.238 + 0.412i)12-s − 1.52·13-s + (−0.202 − 1.48i)14-s + (0.139 + 0.358i)15-s + (0.352 + 0.610i)16-s + (−0.0749 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.36939 - 1.19875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.36939 - 1.19875i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-15.6 + 19.5i)T \) |
| 7 | \( 1 + (-18.5 + 45.3i)T \) |
good | 2 | \( 1 + (-5.18 + 2.99i)T + (8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (1.73 - 2.99i)T + (-40.5 - 70.1i)T^{2} \) |
| 11 | \( 1 + (90.7 - 157. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + 258.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (21.6 - 37.4i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-161. + 93.5i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-374. + 216. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + 647.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (-540. - 312. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-282. + 162. i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 2.88e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.29e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-177. - 307. i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (796. + 459. i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.85e3 + 1.07e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.37e3 - 1.37e3i)T + (6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.44e3 + 1.98e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 7.72e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-931. + 1.61e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (1.99e3 + 3.45e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 - 6.81e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (1.33e3 - 769. i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + 1.29e4T + 8.85e7T^{2} \) |
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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
−15.26069269271568498151578586011, −14.07134702852547499674528733038, −13.08448300655746537008794928948, −12.31699525348064335149736962166, −10.73962630094029329724393488989, −9.837480000545880931864807405974, −7.50098390283616329168875149339, −5.06321930741411014519847624088, −4.59439970290687948182998765093, −2.09614284956682936489680121481,
2.96121635211646215674310572015, 5.24538747299408177026323592946, 6.22067610043553655176518966729, 7.52156744863728778438588471151, 9.646261299389547133975026420392, 11.49568155593995451736777698524, 12.66419425341830412272921757872, 13.71773387798631573231863549259, 14.79173749382159256506276282039, 15.44081494315685262550336799788