| L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.315 + 0.972i)3-s + (0.309 + 0.951i)4-s + (−2.00 − 0.986i)5-s + (0.315 − 0.972i)6-s − 7-s + (0.309 − 0.951i)8-s + (1.58 − 1.14i)9-s + (1.04 + 1.97i)10-s + (−3.56 − 2.59i)11-s + (−0.827 + 0.600i)12-s + (3.29 − 2.39i)13-s + (0.809 + 0.587i)14-s + (0.325 − 2.26i)15-s + (−0.809 + 0.587i)16-s + (−0.486 + 1.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.182 + 0.561i)3-s + (0.154 + 0.475i)4-s + (−0.897 − 0.441i)5-s + (0.128 − 0.396i)6-s − 0.377·7-s + (0.109 − 0.336i)8-s + (0.527 − 0.382i)9-s + (0.330 + 0.625i)10-s + (−1.07 − 0.780i)11-s + (−0.238 + 0.173i)12-s + (0.913 − 0.663i)13-s + (0.216 + 0.157i)14-s + (0.0839 − 0.584i)15-s + (−0.202 + 0.146i)16-s + (−0.118 + 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.490021 - 0.540037i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.490021 - 0.540037i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.00 + 0.986i)T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + (-0.315 - 0.972i)T + (-2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (3.56 + 2.59i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.29 + 2.39i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.486 - 1.49i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.57 + 7.93i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.18 + 1.58i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.424 + 1.30i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.66 + 8.19i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.85 - 1.34i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (5.14 - 3.74i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 + (2.06 + 6.36i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.83 - 5.64i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.62 + 1.90i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (7.99 + 5.80i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.14 - 6.61i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.23 + 6.86i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.01 - 5.82i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-5.24 - 16.1i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.823 + 2.53i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.2 - 8.19i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.18 - 9.79i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.09870768875725162368440778269, −10.36135022664649454990379761392, −9.386877467213652502520304022284, −8.513826984079869164075617964257, −7.83317462510270789327839812629, −6.58418005269257874591172823771, −5.07086606455612834729723882177, −3.86113810036357539998162915690, −2.96940929925434282445124051107, −0.61230496852825593919960455206,
1.75460207956306300705387989646, 3.42573338416744953192247736065, 4.85351107682962102553171473640, 6.30813079617231813607767889714, 7.24477918066438865009143155883, 7.79666545345327595337726631397, 8.656660599816499069602534030737, 10.05938625613209725527593983965, 10.52472155957290253126216139666, 11.78613126329531178046911194468
