| L(s) = 1 | + i·2-s + 3-s − 4-s + 0.561i·5-s + i·6-s − i·8-s + 9-s − 0.561·10-s + 1.43i·11-s − 12-s + (0.561 − 3.56i)13-s + 0.561i·15-s + 16-s − 5.68·17-s + i·18-s − 2.56i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.251i·5-s + 0.408i·6-s − 0.353i·8-s + 0.333·9-s − 0.177·10-s + 0.433i·11-s − 0.288·12-s + (0.155 − 0.987i)13-s + 0.144i·15-s + 0.250·16-s − 1.37·17-s + 0.235i·18-s − 0.587i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.891385741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.891385741\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.561 + 3.56i)T \) |
| good | 5 | \( 1 - 0.561iT - 5T^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 2.56iT - 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 + 10.2iT - 31T^{2} \) |
| 37 | \( 1 + 1.68iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.24iT - 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 12.2iT - 59T^{2} \) |
| 61 | \( 1 + 2.56T + 61T^{2} \) |
| 67 | \( 1 - 7.12iT - 67T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 + 7.43iT - 73T^{2} \) |
| 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
−8.563366398082103737759065738240, −7.60318479517440714835405256655, −7.09681104557620278958306178898, −6.45488267180267305986166985878, −5.47973408174171856314477625130, −4.74954793759031444716002927456, −3.93205851924136161117812890696, −2.98036190595648597365047428641, −2.11524485355977788165632034388, −0.55749121441925118733232411534,
1.13747369718734960942181820643, 2.02052540619842198919323873327, 3.00112053517902904382370006538, 3.72293461044671651501968539682, 4.63692989768248469166880973328, 5.18591537604593243419644402671, 6.51914445050949843614941808123, 6.93010683970260664679317075719, 8.074480249660269224763914009079, 8.831255567902913823023711036572
