L(s) = 1 | + (0.707 + 0.707i)2-s + (1.71 + 0.264i)3-s + 1.00i·4-s + (0.790 + 0.790i)5-s + (1.02 + 1.39i)6-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (2.85 + 0.906i)9-s + 1.11i·10-s + (−3.45 + 3.45i)11-s + (−0.264 + 1.71i)12-s + (−1.18 + 3.40i)13-s − 1.00i·14-s + (1.14 + 1.56i)15-s − 1.00·16-s + 0.401·17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.988 + 0.152i)3-s + 0.500i·4-s + (0.353 + 0.353i)5-s + (0.417 + 0.570i)6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.953 + 0.302i)9-s + 0.353i·10-s + (−1.04 + 1.04i)11-s + (−0.0764 + 0.494i)12-s + (−0.329 + 0.944i)13-s − 0.267i·14-s + (0.295 + 0.403i)15-s − 0.250·16-s + 0.0972·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98598 + 1.63186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98598 + 1.63186i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-1.71 - 0.264i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (1.18 - 3.40i)T \) |
good | 5 | \( 1 + (-0.790 - 0.790i)T + 5iT^{2} \) |
| 11 | \( 1 + (3.45 - 3.45i)T - 11iT^{2} \) |
| 17 | \( 1 - 0.401T + 17T^{2} \) |
| 19 | \( 1 + (-4.88 + 4.88i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.12T + 23T^{2} \) |
| 29 | \( 1 + 1.16iT - 29T^{2} \) |
| 31 | \( 1 + (-2.88 + 2.88i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.48 + 4.48i)T + 37iT^{2} \) |
| 41 | \( 1 + (6.76 + 6.76i)T + 41iT^{2} \) |
| 43 | \( 1 - 2.74iT - 43T^{2} \) |
| 47 | \( 1 + (5.67 - 5.67i)T - 47iT^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 + (-8.03 + 8.03i)T - 59iT^{2} \) |
| 61 | \( 1 + 0.717T + 61T^{2} \) |
| 67 | \( 1 + (0.693 - 0.693i)T - 67iT^{2} \) |
| 71 | \( 1 + (5.15 + 5.15i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.0455 + 0.0455i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + (-1.74 - 1.74i)T + 83iT^{2} \) |
| 89 | \( 1 + (8.55 - 8.55i)T - 89iT^{2} \) |
| 97 | \( 1 + (7.03 - 7.03i)T - 97iT^{2} \) |
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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
−10.86496740535798368783275545333, −9.863981665960822305058771935211, −9.313906420314610709072309106744, −8.188354148951237081587444586029, −7.17503259546394226017878793944, −6.83977687288458273982954020764, −5.18925938479856116207685114723, −4.44281074535180171222190952448, −3.13134788810275943064210700172, −2.22643909446892652218646148661,
1.32279090456245854732054618603, 2.93042233977730419264481578599, 3.32685658463200025243320472127, 5.03065428853143995867298088418, 5.67630596998430516594579295486, 7.06608802623152177105484032640, 8.126100563554320547901905955900, 8.820101059826930105735153325583, 9.909802656240178326432470497826, 10.37326034259748948483754223670