| L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.973 − 2.99i)3-s + (0.309 − 0.951i)4-s + (−2.20 − 0.393i)5-s + (0.973 + 2.99i)6-s − 2.52·7-s + (0.309 + 0.951i)8-s + (−5.59 − 4.06i)9-s + (2.01 − 0.975i)10-s + (−1.97 + 1.43i)11-s + (−2.54 − 1.85i)12-s + (2.06 + 1.50i)13-s + (2.04 − 1.48i)14-s + (−3.32 + 6.21i)15-s + (−0.809 − 0.587i)16-s + (0.368 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.561 − 1.72i)3-s + (0.154 − 0.475i)4-s + (−0.984 − 0.175i)5-s + (0.397 + 1.22i)6-s − 0.954·7-s + (0.109 + 0.336i)8-s + (−1.86 − 1.35i)9-s + (0.636 − 0.308i)10-s + (−0.595 + 0.432i)11-s + (−0.735 − 0.534i)12-s + (0.573 + 0.416i)13-s + (0.546 − 0.396i)14-s + (−0.857 + 1.60i)15-s + (−0.202 − 0.146i)16-s + (0.0894 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0858 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0858 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0669227 + 0.0729404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0669227 + 0.0729404i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.20 + 0.393i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
| good | 3 | \( 1 + (-0.973 + 2.99i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + (1.97 - 1.43i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.06 - 1.50i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.368 - 1.13i)T + (-13.7 + 9.99i)T^{2} \) |
| 23 | \( 1 + (-0.193 + 0.140i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.625 - 1.92i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.52 - 4.69i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.849 - 0.617i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (9.79 + 7.11i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 + (-0.00246 + 0.00758i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (1.04 - 3.22i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (1.92 + 1.40i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-10.4 + 7.62i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.44 - 13.6i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (2.22 - 6.86i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (8.30 - 6.03i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.15 + 12.7i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.22 - 13.0i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (9.92 - 7.20i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.06 + 6.36i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.10137160072624915124319769240, −8.958058405760604184542602808627, −8.447315609149425638201603560963, −7.73691597281292823630686473902, −6.88073325886710774377584532962, −6.61894817118440482623408628551, −5.34757995661699595845920744880, −3.69337207771541199172110391979, −2.64479433635070099618591897960, −1.31809687102124579420488153970,
0.05425417436105500803338077709, 2.79869788010502328085743976451, 3.37028272256274200860332966422, 4.08534123137919698065393698858, 5.16506865350980332044331026441, 6.40181742476825133320195395624, 7.80216227041733080359082699614, 8.335799153962286168840570812884, 9.132230405684420456915666455680, 9.916142783406400500963659776587
