| L(s) = 1 | − i·2-s + 3-s − 4-s + 3.56i·5-s − i·6-s + i·8-s + 9-s + 3.56·10-s − 5.56i·11-s − 12-s + (−3.56 − 0.561i)13-s + 3.56i·15-s + 16-s + 6.68·17-s − i·18-s − 1.56i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.59i·5-s − 0.408i·6-s + 0.353i·8-s + 0.333·9-s + 1.12·10-s − 1.67i·11-s − 0.288·12-s + (−0.987 − 0.155i)13-s + 0.919i·15-s + 0.250·16-s + 1.62·17-s − 0.235i·18-s − 0.358i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682855116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682855116\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (3.56 + 0.561i)T \) |
| good | 5 | \( 1 - 3.56iT - 5T^{2} \) |
| 11 | \( 1 + 5.56iT - 11T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 + 1.56iT - 19T^{2} \) |
| 23 | \( 1 + 6.68T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 + 6.24iT - 31T^{2} \) |
| 37 | \( 1 + 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 + 6.43T + 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 4.24iT - 59T^{2} \) |
| 61 | \( 1 - 1.56T + 61T^{2} \) |
| 67 | \( 1 - 1.12iT - 67T^{2} \) |
| 71 | \( 1 + 9.36iT - 71T^{2} \) |
| 73 | \( 1 - 11.5iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.153929001370483080499805326518, −7.75609812728387995574094991631, −6.92937865812085036670730237750, −5.98791276603164938453349139854, −5.40487244296739256636806767877, −3.97950747940066417667162600432, −3.41691938851331204841228803633, −2.79491089641858014169331483239, −2.07473122963751526216263115121, −0.47482386793054703418209224041,
1.19603627635854026925673851246, 2.04978226664106756026810346045, 3.44456631299192908983119225726, 4.46435379890749963783489388658, 4.85071078499621171806624781608, 5.52851085819616010763122175276, 6.59982345674266882348712105583, 7.46858491167416722637369952625, 7.969234938022885172236928009486, 8.494139106446630623792255352955
