L(s) = 1 | + (1.83 − 0.596i)2-s + (−0.304 + 0.418i)3-s + (1.40 − 1.01i)4-s + (−1.88 + 1.21i)5-s + (−0.309 + 0.951i)6-s + (0.526 + 0.725i)7-s + (−0.304 + 0.418i)8-s + (0.844 + 2.59i)9-s + (−2.73 + 3.34i)10-s + 0.896i·12-s + (4.03 − 1.31i)13-s + (1.40 + 1.01i)14-s + (0.0652 − 1.15i)15-s + (−1.37 + 4.24i)16-s + (−0.984 − 0.319i)17-s + (3.10 + 4.26i)18-s + ⋯ |
L(s) = 1 | + (1.29 − 0.422i)2-s + (−0.175 + 0.241i)3-s + (0.700 − 0.509i)4-s + (−0.840 + 0.541i)5-s + (−0.126 + 0.388i)6-s + (0.199 + 0.274i)7-s + (−0.107 + 0.148i)8-s + (0.281 + 0.866i)9-s + (−0.863 + 1.05i)10-s + 0.258i·12-s + (1.11 − 0.363i)13-s + (0.374 + 0.272i)14-s + (0.0168 − 0.298i)15-s + (−0.344 + 1.06i)16-s + (−0.238 − 0.0775i)17-s + (0.731 + 1.00i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.08771 + 1.01888i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08771 + 1.01888i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.88 - 1.21i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.83 + 0.596i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.304 - 0.418i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (-0.526 - 0.725i)T + (-2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-4.03 + 1.31i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.984 + 0.319i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.01 - 3.64i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 6.31iT - 23T^{2} \) |
| 29 | \( 1 + (5.60 - 4.07i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.62 + 5.01i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.93 + 5.41i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.40 - 1.01i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (-2.41 + 3.32i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-11.3 + 3.69i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.82 - 2.78i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (2.55 - 7.86i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 14.9iT - 67T^{2} \) |
| 71 | \( 1 + (-0.678 + 2.08i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.87 - 3.96i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.68 - 5.19i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-9.41 - 3.05i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 + (0.624 - 0.202i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.03275825598595688588285719812, −10.42042105475290226138042833374, −9.025153415653962190223674588118, −7.968919814040207556571894315663, −7.17741229774607467262466705918, −5.71743705009923516977817147788, −5.26475108666275402840636782919, −3.89317294813209442706152370859, −3.51245818614825957346546831934, −2.04023954911975265092074805484,
0.945898997790053654127687062827, 3.22699145875864234625970480723, 4.09157458585711805890532828110, 4.79840430954998626891830831016, 5.93042751964358667564503310274, 6.75517566781960620857568607020, 7.52392482164961813333773790968, 8.719183817250321768876680720878, 9.478536666718701233857112565723, 10.94427882441207673773794295187