| L(s) = 1 | + i·3-s − 4.10i·5-s − 7-s − 9-s − 2.67i·11-s + 3.02i·13-s + 4.10·15-s − 5.12·17-s − 2.78i·19-s − i·21-s − 7.12·23-s − 11.8·25-s − i·27-s − 8.83i·29-s + 1.42·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.83i·5-s − 0.377·7-s − 0.333·9-s − 0.807i·11-s + 0.838i·13-s + 1.05·15-s − 1.24·17-s − 0.637i·19-s − 0.218i·21-s − 1.48·23-s − 2.36·25-s − 0.192i·27-s − 1.63i·29-s + 0.255·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.424977 - 0.771859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.424977 - 0.771859i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 4.10iT - 5T^{2} \) |
| 11 | \( 1 + 2.67iT - 11T^{2} \) |
| 13 | \( 1 - 3.02iT - 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 2.78iT - 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 + 8.83iT - 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 - 1.42iT - 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 2.39iT - 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 + 2.78iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 5.17iT - 61T^{2} \) |
| 67 | \( 1 - 0.244iT - 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 + 6.25T + 79T^{2} \) |
| 83 | \( 1 - 9.35iT - 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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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
−9.956433821050604516319232047115, −9.231594123215014726753538795353, −8.705118932347589626170021762245, −7.921818869765924915371848172919, −6.40007635680538998087198978324, −5.58547812122405577929071209886, −4.49016462901033389686382288489, −4.00408799469641709478310877918, −2.19521221554476824168278841791, −0.44227034401757427938518163838,
2.09570872080927239266768447886, 2.97535289615267483865543556691, 4.08042045884049859012933667854, 5.75156370564895031121058169434, 6.49324682823592232524772693385, 7.22751575484445746012354934696, 7.85950871581726271120794036372, 9.131954189525344328891188643485, 10.31096834716367268387712005226, 10.55943424714475272906171880340
