| 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
−9.038023378071371266602837294597, −7.926796371280011190750072448717, −7.47401569228022794264622549798, −6.59161974624340901904239326845, −5.67430837944134285475727722104, −4.83924936638250761667998494933, −3.87297923137974952024021335531, −2.60146538159682469564358868507, −1.42847262390158066641769089624, −0.60149080567802404059670583758,
1.37014103510169540218374347870, 2.78749455983967703290618387114, 3.60395251008902577842369193434, 4.50839864523087671646641965159, 5.26769914466534657477871017086, 6.53122164346171314155753109441, 6.95204913230674060179786951127, 7.943955825936049251176153144345, 9.181050326514637730968141710891, 9.338379144402214228985031480070
