| L(s) = 1 | + (2.93 − 2.72i)2-s + (0.0921 − 0.0138i)3-s + (1.19 − 15.9i)4-s + (−11.3 + 28.9i)5-s + (0.232 − 0.291i)6-s + (70.0 + 109. i)7-s + (−39.9 − 50.0i)8-s + (−232. + 71.6i)9-s + (45.4 + 115. i)10-s + (−542. − 167. i)11-s + (−0.111 − 1.48i)12-s + (32.9 + 144. i)13-s + (502. + 129. i)14-s + (−0.644 + 2.82i)15-s + (−253. − 38.1i)16-s + (−1.07e3 + 733. i)17-s + ⋯ |
| L(s) = 1 | + (0.518 − 0.480i)2-s + (0.00591 − 0.000891i)3-s + (0.0373 − 0.498i)4-s + (−0.203 + 0.517i)5-s + (0.00263 − 0.00330i)6-s + (0.540 + 0.841i)7-s + (−0.220 − 0.276i)8-s + (−0.955 + 0.294i)9-s + (0.143 + 0.366i)10-s + (−1.35 − 0.416i)11-s + (−0.000223 − 0.00298i)12-s + (0.0540 + 0.236i)13-s + (0.684 + 0.175i)14-s + (−0.000740 + 0.00324i)15-s + (−0.247 − 0.0372i)16-s + (−0.902 + 0.615i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.719743 + 0.891943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.719743 + 0.891943i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.93 + 2.72i)T \) |
| 7 | \( 1 + (-70.0 - 109. i)T \) |
| good | 3 | \( 1 + (-0.0921 + 0.0138i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (11.3 - 28.9i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (542. + 167. i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (-32.9 - 144. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (1.07e3 - 733. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-1.22e3 - 2.11e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (164. + 112. i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (2.73e3 + 1.31e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (705. - 1.22e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-565. - 7.53e3i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (6.11e3 + 7.67e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-4.58e3 + 5.75e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-5.30e3 + 4.91e3i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (964. - 1.28e4i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (4.34e3 + 1.10e4i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (3.45e3 + 4.61e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (2.08e4 - 3.60e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-1.45e4 + 7.01e3i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (2.16e4 + 2.00e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (3.44e4 + 5.96e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (8.16e3 - 3.57e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-1.17e5 + 3.61e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 - 1.02e5T + 8.58e9T^{2} \) |
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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
−13.28140513986488674823208788090, −12.08465789856506946269981490069, −11.20041727881782935668960391802, −10.44685544102776534501203984167, −8.835305347846702068658401604769, −7.79837233125507271036763517120, −6.00362689280056381871988253574, −5.10989192371024953808984575752, −3.28990486659644225896846270077, −2.11369005157439586706349383448,
0.35451214716469725633764696547, 2.77926850247802801797687377307, 4.51779266745211603965773280990, 5.39117245975629963266548155271, 7.05033383698533848401479901659, 8.002412067255688161884680582023, 9.129752758519934512794724196238, 10.76358979128055115160784497498, 11.64681594433244518606419039424, 12.96718238623208775108952082532
