| L(s) = 1 | + (−0.932 + 1.06i)2-s + (−1.73 + 0.0711i)3-s + (−0.261 − 1.98i)4-s + (−1.84 − 1.25i)5-s + (1.53 − 1.90i)6-s − 2.80·7-s + (2.35 + 1.57i)8-s + (2.98 − 0.246i)9-s + (3.06 − 0.792i)10-s + (1.50 + 1.08i)11-s + (0.593 + 3.41i)12-s + (2.01 + 2.76i)13-s + (2.61 − 2.98i)14-s + (3.28 + 2.04i)15-s + (−3.86 + 1.03i)16-s + (0.345 − 1.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.659 + 0.751i)2-s + (−0.999 + 0.0410i)3-s + (−0.130 − 0.991i)4-s + (−0.826 − 0.562i)5-s + (0.627 − 0.778i)6-s − 1.06·7-s + (0.831 + 0.555i)8-s + (0.996 − 0.0820i)9-s + (0.968 − 0.250i)10-s + (0.452 + 0.328i)11-s + (0.171 + 0.985i)12-s + (0.558 + 0.768i)13-s + (0.698 − 0.797i)14-s + (0.849 + 0.528i)15-s + (−0.965 + 0.258i)16-s + (0.0837 − 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.435945 + 0.252642i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.435945 + 0.252642i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.932 - 1.06i)T \) |
| 3 | \( 1 + (1.73 - 0.0711i)T \) |
| 5 | \( 1 + (1.84 + 1.25i)T \) |
| good | 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 + (-1.50 - 1.08i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.01 - 2.76i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.345 + 1.06i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-7.11 - 2.31i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.23 + 1.70i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.56 + 1.15i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (8.53 + 2.77i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.97 - 6.84i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.81 - 5.25i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 + (-0.999 + 0.324i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.595 + 1.83i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.91 + 5.02i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.730 + 0.530i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.462 - 1.42i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.11 - 3.41i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.45 + 10.2i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.76 + 1.54i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.84 - 1.89i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (9.96 - 13.7i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (16.7 - 5.45i)T + (78.4 - 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.74042115241764964677992000677, −11.00345167926183254664048708497, −9.689463510161067672648587166634, −9.302970570875401251419278877227, −7.88610090606810397814393532789, −6.98570359945694120915960028678, −6.17084502421668397102124093722, −5.06690356823620828241587290294, −3.91724953767021900165821109031, −1.00955077392731269012652823155,
0.73572821508964950110782005188, 3.10529342167305674255327557203, 3.97338427065597010099585925112, 5.68454090996415355556668012105, 6.95173507989031908741168590570, 7.58854780469318060587334549687, 9.003975747681947021113270745435, 9.920007903901167315687529343541, 10.88990794069882594261737713921, 11.31275261295130143368233863668
