| L(s) = 1 | − i·2-s − 4-s − 1.41i·5-s + (−2.12 + 1.58i)7-s + i·8-s − 1.41·10-s + (3.16 + i)11-s − 1.41·13-s + (1.58 + 2.12i)14-s + 16-s + 4.47·17-s + 2.82·19-s + 1.41i·20-s + (1 − 3.16i)22-s − 3.16·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.632i·5-s + (−0.801 + 0.597i)7-s + 0.353i·8-s − 0.447·10-s + (0.953 + 0.301i)11-s − 0.392·13-s + (0.422 + 0.566i)14-s + 0.250·16-s + 1.08·17-s + 0.648·19-s + 0.316i·20-s + (0.213 − 0.674i)22-s − 0.659·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494291864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494291864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.12 - 1.58i)T \) |
| 11 | \( 1 + (-3.16 - i)T \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 8.94iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 + 6.32iT - 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 + 2.82iT - 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 3.16iT - 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 - 16.9iT - 89T^{2} \) |
| 97 | \( 1 + 4.47iT - 97T^{2} \) |
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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
−9.489382926494019164746242683875, −8.929598308420662417434800473616, −7.969386677222384078315034055196, −6.99019899429480685929631710994, −5.91508564786195742236592272398, −5.19964337766802914052882995816, −4.11000140510819991918139715694, −3.27315210999057937922649719885, −2.13550477322461214444061761278, −0.831414118950432119625578183560,
1.00644490282406680974390549487, 2.92643753530745311714825785173, 3.69224167881291866823620905194, 4.68495841262077689317275998844, 5.93400058659106275239574608056, 6.42437203381782185026599339773, 7.30201760383178441400625749596, 7.83147221560541162401190727356, 8.998512650608420334967800377191, 9.690377841978117545909582462613
