| L(s) = 1 | + (0.707 − 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (−1.24 − 0.717i)5-s − 2.44i·6-s + (1.74 + 6.77i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (−1.75 + 1.01i)10-s + (−3 − 5.19i)11-s + (−2.99 − 1.73i)12-s + 21.3i·13-s + (9.53 + 2.65i)14-s − 2.48·15-s + (−2.00 + 3.46i)16-s + (−7.75 + 4.47i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.248 − 0.143i)5-s − 0.408i·6-s + (0.248 + 0.968i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.175 + 0.101i)10-s + (−0.272 − 0.472i)11-s + (−0.249 − 0.144i)12-s + 1.64i·13-s + (0.681 + 0.189i)14-s − 0.165·15-s + (−0.125 + 0.216i)16-s + (−0.456 + 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.21199 - 0.602257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21199 - 0.602257i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.74 - 6.77i)T \) |
| good | 5 | \( 1 + (1.24 + 0.717i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 21.3iT - 169T^{2} \) |
| 17 | \( 1 + (7.75 - 4.47i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (6.25 + 3.61i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-18.7 + 32.4i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 33.9T + 841T^{2} \) |
| 31 | \( 1 + (-38.2 + 22.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-13.9 + 24.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 54.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.48T + 1.84e3T^{2} \) |
| 47 | \( 1 + (37.2 + 21.5i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-42.7 - 74.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (35.6 - 20.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.02 + 0.594i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (2.19 + 3.80i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 137.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-68.3 + 39.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (49.1 - 85.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (18 + 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 10.9iT - 9.40e3T^{2} \) |
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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
−15.32746647442409093298588963826, −14.36868209313906445850573605914, −13.22394117790519796923113074819, −12.10140574466711341635237200530, −11.12821572458445703318137204726, −9.346296849718374923314698171812, −8.364627548432850734657708634602, −6.35902541952197033626076331665, −4.42902515379082659072706649281, −2.36411800528140250691883631837,
3.52432439487018561702569581882, 5.15068531751407392523279398199, 7.20074082677624703085269624357, 8.118739712697830033142099782363, 9.815225387421475556026287969762, 11.09659642757673927786941720914, 12.89615350901325075273758591133, 13.71787695716482870216919478307, 15.05082933801931836451444588009, 15.58040436703246579004146548584
