L(s) = 1 | + (0.227 + 1.71i)3-s + (−2.89 + 0.781i)9-s − 4.88·13-s + 4.79i·23-s − 5·25-s + (−1.99 − 4.79i)27-s + 8.43i·29-s − 11.1·31-s + (−1.11 − 8.38i)39-s − 12.1i·41-s + 6.55i·47-s + 7·49-s + 9.59i·59-s + (−8.23 + 1.09i)69-s − 14.0i·71-s + ⋯ |
L(s) = 1 | + (0.131 + 0.991i)3-s + (−0.965 + 0.260i)9-s − 1.35·13-s + 0.999i·23-s − 25-s + (−0.384 − 0.922i)27-s + 1.56i·29-s − 1.99·31-s + (−0.177 − 1.34i)39-s − 1.90i·41-s + 0.956i·47-s + 49-s + 1.24i·59-s + (−0.991 + 0.131i)69-s − 1.66i·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5822248323\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5822248323\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.227 - 1.71i)T \) |
| 23 | \( 1 - 4.79iT \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 29 | \( 1 - 8.43iT - 29T^{2} \) |
| 31 | \( 1 + 11.1T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 12.1iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 6.55iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.59iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 14.0iT - 71T^{2} \) |
| 73 | \( 1 + 7.61T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.31460126332418381481319789702, −9.291277829829417282086688204303, −9.046666611451662801801660511963, −7.74140199427560032426815789422, −7.15302093740207360358452405998, −5.70268171543729627810051109192, −5.19021625424415694343189818999, −4.12178222353542567526035301681, −3.26422480426502116050768424060, −2.06424180663943459894146395132,
0.23108825195054124953272846178, 1.91236311577849384470233180245, 2.75547565064089957888725521802, 4.08468746415526629377346927125, 5.26908461416374846032776127389, 6.12853900731327688339821405110, 7.04181553181393912786648622476, 7.68980615441576677553231994716, 8.428389121194619065190399025298, 9.417136332996148862334366753997