| L(s) = 1 | + (1.35 + 0.388i)2-s + (0.819 − 0.573i)3-s + (1.69 + 1.05i)4-s + (−3.73 + 0.326i)5-s + (1.33 − 0.462i)6-s + (1.21 + 2.10i)7-s + (1.90 + 2.09i)8-s + (0.342 − 0.939i)9-s + (−5.20 − 1.00i)10-s + (1.16 − 0.313i)11-s + (1.99 − 0.109i)12-s + (5.16 + 3.61i)13-s + (0.836 + 3.33i)14-s + (−2.87 + 2.41i)15-s + (1.77 + 3.58i)16-s + (−5.89 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (0.961 + 0.274i)2-s + (0.472 − 0.331i)3-s + (0.849 + 0.527i)4-s + (−1.67 + 0.146i)5-s + (0.545 − 0.188i)6-s + (0.459 + 0.796i)7-s + (0.672 + 0.740i)8-s + (0.114 − 0.313i)9-s + (−1.64 − 0.317i)10-s + (0.352 − 0.0943i)11-s + (0.576 − 0.0317i)12-s + (1.43 + 1.00i)13-s + (0.223 + 0.892i)14-s + (−0.741 + 0.622i)15-s + (0.443 + 0.896i)16-s + (−1.42 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.22145 + 1.54971i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.22145 + 1.54971i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 - 0.388i)T \) |
| 3 | \( 1 + (-0.819 + 0.573i)T \) |
| 19 | \( 1 + (3.38 - 2.74i)T \) |
| good | 5 | \( 1 + (3.73 - 0.326i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.21 - 2.10i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.16 + 0.313i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-5.16 - 3.61i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (5.89 - 2.14i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.31 + 1.10i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.86 + 3.66i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (1.49 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.36 - 3.36i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.111 - 0.633i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.537 - 6.14i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (1.22 - 3.35i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (7.20 + 0.629i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-2.47 + 5.30i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-6.47 - 0.566i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (2.28 + 4.90i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-5.39 + 6.42i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (9.67 + 1.70i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.72 + 15.4i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (6.39 + 1.71i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (1.79 + 10.1i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (4.11 + 11.3i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.72276217066589349462926535539, −8.878635326122302945283656290936, −8.383461389771848782160543016058, −7.81900441215509021031173052262, −6.57498145329155922161298217452, −6.27207039379464813984611177865, −4.50759449493161179880624039380, −4.14147502983587233917313314140, −3.10925722419514560181658176159, −1.84159271747637096761375797087,
0.963862242365187728952985317881, 2.79847206869051635370864133748, 3.84657252843106177986370028710, 4.23593094502131881263071540426, 5.11960260443715723440262257815, 6.64133385631856777580090500525, 7.27233380741886363380765202747, 8.263900134372190232352431056449, 8.847566768849109137967632111697, 10.37796691002711570240050627879
