| L(s) = 1 | + (1.53 − 0.795i)3-s − 3.80·5-s + (1.73 − 2.44i)9-s + 0.357i·11-s − 4.04i·13-s + (−5.84 + 3.02i)15-s − 0.103·17-s + 2.45i·19-s + 1.33i·23-s + 9.44·25-s + (0.723 − 5.14i)27-s + 4.97i·29-s − 7.88i·31-s + (0.284 + 0.549i)33-s − 10.9·37-s + ⋯ |
| L(s) = 1 | + (0.888 − 0.459i)3-s − 1.69·5-s + (0.578 − 0.815i)9-s + 0.107i·11-s − 1.12i·13-s + (−1.50 + 0.780i)15-s − 0.0252·17-s + 0.563i·19-s + 0.277i·23-s + 1.88·25-s + (0.139 − 0.990i)27-s + 0.923i·29-s − 1.41i·31-s + (0.0494 + 0.0957i)33-s − 1.79·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3528298585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3528298585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.53 + 0.795i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.80T + 5T^{2} \) |
| 11 | \( 1 - 0.357iT - 11T^{2} \) |
| 13 | \( 1 + 4.04iT - 13T^{2} \) |
| 17 | \( 1 + 0.103T + 17T^{2} \) |
| 19 | \( 1 - 2.45iT - 19T^{2} \) |
| 23 | \( 1 - 1.33iT - 23T^{2} \) |
| 29 | \( 1 - 4.97iT - 29T^{2} \) |
| 31 | \( 1 + 7.88iT - 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6.15T + 41T^{2} \) |
| 43 | \( 1 + 0.502T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 5.86iT - 53T^{2} \) |
| 59 | \( 1 + 7.54T + 59T^{2} \) |
| 61 | \( 1 - 9.47iT - 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 + 5.78iT - 71T^{2} \) |
| 73 | \( 1 - 0.235iT - 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 + 9.07T + 83T^{2} \) |
| 89 | \( 1 - 6.82T + 89T^{2} \) |
| 97 | \( 1 - 5.14iT - 97T^{2} \) |
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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
−8.367472640402482638530685145797, −7.82801489366134883832690240738, −7.34855406212314647362085870932, −6.52409384112053871609453644181, −5.31184344327139376178375990693, −4.27003346775372383316295177116, −3.50505220402099550821233810000, −2.96025436256483676389208896459, −1.49287493733244921066351227714, −0.10655647260328114732855199197,
1.74152896098541145829522396347, 3.07589323596538301084994410236, 3.65924214437667077977318092556, 4.48547005203884958840169155949, 5.02658477140998865326466110821, 6.72011672723025846675097647204, 7.12360778379148876367653426316, 8.114479191415330473529751248077, 8.489041040141881928884112140525, 9.182677035435430885622078665605
