L(s) = 1 | + 2.54i·3-s + 4.11i·5-s + 2.54·9-s + 3.26·11-s + 5.88i·13-s − 10.4·15-s + 13.8i·17-s − 15.8i·19-s + 36.5·23-s + 8.03·25-s + 29.3i·27-s − 28.4·29-s + 41.8i·31-s + 8.30i·33-s + 14.2·37-s + ⋯ |
L(s) = 1 | + 0.847i·3-s + 0.823i·5-s + 0.282·9-s + 0.297·11-s + 0.452i·13-s − 0.697·15-s + 0.816i·17-s − 0.832i·19-s + 1.58·23-s + 0.321·25-s + 1.08i·27-s − 0.981·29-s + 1.34i·31-s + 0.251i·33-s + 0.386·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.006191964\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.006191964\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.54iT - 9T^{2} \) |
| 5 | \( 1 - 4.11iT - 25T^{2} \) |
| 11 | \( 1 - 3.26T + 121T^{2} \) |
| 13 | \( 1 - 5.88iT - 169T^{2} \) |
| 17 | \( 1 - 13.8iT - 289T^{2} \) |
| 19 | \( 1 + 15.8iT - 361T^{2} \) |
| 23 | \( 1 - 36.5T + 529T^{2} \) |
| 29 | \( 1 + 28.4T + 841T^{2} \) |
| 31 | \( 1 - 41.8iT - 961T^{2} \) |
| 37 | \( 1 - 14.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 33.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 84.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 67.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 29.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 54.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 83.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 125. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 70.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 27.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 11.3iT - 9.40e3T^{2} \) |
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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
−9.338379144402214228985031480070, −9.181050326514637730968141710891, −7.943955825936049251176153144345, −6.95204913230674060179786951127, −6.53122164346171314155753109441, −5.26769914466534657477871017086, −4.50839864523087671646641965159, −3.60395251008902577842369193434, −2.78749455983967703290618387114, −1.37014103510169540218374347870,
0.60149080567802404059670583758, 1.42847262390158066641769089624, 2.60146538159682469564358868507, 3.87297923137974952024021335531, 4.83924936638250761667998494933, 5.67430837944134285475727722104, 6.59161974624340901904239326845, 7.47401569228022794264622549798, 7.926796371280011190750072448717, 9.038023378071371266602837294597