L(s) = 1 | − 1.73·3-s + 2.44i·5-s + 4.24i·7-s + 2.99·9-s − 3.46·11-s − 13-s − 4.24i·15-s − 4.89i·17-s − 4.24i·19-s − 7.34i·21-s − 3.46·23-s − 0.999·25-s − 5.19·27-s + 9.79i·29-s + 4.24i·31-s + ⋯ |
L(s) = 1 | − 1.00·3-s + 1.09i·5-s + 1.60i·7-s + 0.999·9-s − 1.04·11-s − 0.277·13-s − 1.09i·15-s − 1.18i·17-s − 0.973i·19-s − 1.60i·21-s − 0.722·23-s − 0.199·25-s − 1.00·27-s + 1.81i·29-s + 0.762i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(-0.459232i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.459232i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 9.79iT - 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 2.44iT - 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 9.79iT - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 7.34iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.00464890092503881929624088513, −10.40416239142399694598481450391, −9.454103821536623106573140231276, −8.464702140738351302091085519155, −7.13235680997587725242373085688, −6.68780563970446105807940134854, −5.34325655223371026628825982952, −5.13149114401127276965856839506, −3.18242013980329242550049855596, −2.24219013437763213611978408752,
0.28056991592860437408476689901, 1.60703190807228123628453084905, 3.90855415044564565245888732452, 4.54176220760005337175147375345, 5.54951168501505297692288792841, 6.44001765553501408662500086135, 7.69239149471439162455635689422, 8.084259174601923980337687535918, 9.636393088347292904625259125140, 10.31305931648574971793856561759