| L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.924 − 2.84i)3-s + (0.309 + 0.951i)4-s + (2.23 + 0.0110i)5-s + (−0.924 + 2.84i)6-s + 2.08·7-s + (0.309 − 0.951i)8-s + (−4.81 + 3.49i)9-s + (−1.80 − 1.32i)10-s + (3.40 + 2.47i)11-s + (2.42 − 1.75i)12-s + (−0.805 + 0.585i)13-s + (−1.68 − 1.22i)14-s + (−2.03 − 6.37i)15-s + (−0.809 + 0.587i)16-s + (1.62 − 4.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.533 − 1.64i)3-s + (0.154 + 0.475i)4-s + (0.999 + 0.00495i)5-s + (−0.377 + 1.16i)6-s + 0.788·7-s + (0.109 − 0.336i)8-s + (−1.60 + 1.16i)9-s + (−0.569 − 0.418i)10-s + (1.02 + 0.744i)11-s + (0.698 − 0.507i)12-s + (−0.223 + 0.162i)13-s + (−0.451 − 0.327i)14-s + (−0.525 − 1.64i)15-s + (−0.202 + 0.146i)16-s + (0.393 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.723935 - 1.15332i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.723935 - 1.15332i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-2.23 - 0.0110i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| good | 3 | \( 1 + (0.924 + 2.84i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 + (-3.40 - 2.47i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.805 - 0.585i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.62 + 4.99i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (-2.99 - 2.17i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.246 - 0.759i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.42 + 7.46i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.69 + 5.58i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.680 + 0.494i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.529T + 43T^{2} \) |
| 47 | \( 1 + (-1.35 - 4.15i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.04 + 6.30i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (11.0 - 8.05i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.874 - 0.635i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.37 - 7.30i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.68 + 5.19i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (7.09 + 5.15i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.07 - 6.37i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.75 + 11.5i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (2.61 + 1.90i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.0530 - 0.163i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.604632117718119849777326776640, −9.084898646443395189901569139907, −7.84543297353837923817144575380, −7.33356586600950213973321566282, −6.52184161916154652671762905127, −5.70117843687592818058926613090, −4.60302200286826411866317338501, −2.66408167515787204818917451837, −1.79325492718382810295682948030, −1.00802517033471778355203377203,
1.31603919278927903713702518604, 3.09533944097027746027680941248, 4.38276012541457281116586009487, 5.12148101424363946455219483167, 5.97028217002208764638949637398, 6.57038708740270577312940368083, 8.171664031915059812055886759723, 8.865506451116678382485757783360, 9.497490029754767796852590968277, 10.26728644442640412033315112563
