| L(s) = 1 | − i·2-s + (−0.420 − 1.68i)3-s − 4-s + (−1.68 + 0.420i)6-s + (−2.57 − 0.595i)7-s + i·8-s + (−2.64 + 1.41i)9-s + 2.82i·11-s + (0.420 + 1.68i)12-s + 3.36i·13-s + (−0.595 + 2.57i)14-s + 16-s + 4.75·17-s + (1.41 + 2.64i)18-s + 5.59i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.242 − 0.970i)3-s − 0.5·4-s + (−0.685 + 0.171i)6-s + (−0.974 − 0.224i)7-s + 0.353i·8-s + (−0.881 + 0.471i)9-s + 0.852i·11-s + (0.121 + 0.485i)12-s + 0.931i·13-s + (−0.159 + 0.688i)14-s + 0.250·16-s + 1.15·17-s + (0.333 + 0.623i)18-s + 1.28i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8260233680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8260233680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.420 + 1.68i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.57 + 0.595i)T \) |
| good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 3.36iT - 13T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 - 5.59iT - 19T^{2} \) |
| 23 | \( 1 + 7.29iT - 23T^{2} \) |
| 29 | \( 1 - 0.500iT - 29T^{2} \) |
| 31 | \( 1 - 3.06iT - 31T^{2} \) |
| 37 | \( 1 - 3.32T + 37T^{2} \) |
| 41 | \( 1 + 4.33T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 7.82T + 47T^{2} \) |
| 53 | \( 1 - 8.58iT - 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 - 2.52iT - 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 9.81iT - 71T^{2} \) |
| 73 | \( 1 - 5.53iT - 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 - 6.97T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−10.11706156814912327050389514448, −9.189582339320792947647666979492, −8.277869292671844945149170111233, −7.34787823088684172538699403121, −6.58454568702410857084609060653, −5.75820983970723436318768687574, −4.56222933212755217843928073111, −3.44828216670138822155682437470, −2.35881035016493193817386677824, −1.23389109159400260357459850646,
0.42768127843963389933949620909, 3.09034271373556425543171474260, 3.56484600365987413967235489540, 4.95221803447386323392869553407, 5.65966055666434786446350979885, 6.26530088889604187217996927510, 7.40587929556632083802642008127, 8.341377478264652838684005818109, 9.183181771124693380806659090790, 9.770782670819329324588431198359
