| L(s) = 1 | + i·3-s + i·5-s − 5.01·7-s − 9-s − 2.36·11-s + 5.01·13-s − 15-s + 1.91i·17-s + 4.78·19-s − 5.01i·21-s + (3.28 − 3.49i)23-s − 25-s − i·27-s + 5.75·29-s + 3.94i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447i·5-s − 1.89·7-s − 0.333·9-s − 0.713·11-s + 1.39·13-s − 0.258·15-s + 0.464i·17-s + 1.09·19-s − 1.09i·21-s + (0.684 − 0.728i)23-s − 0.200·25-s − 0.192i·27-s + 1.06·29-s + 0.708i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.251360600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.251360600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-3.28 + 3.49i)T \) |
| good | 7 | \( 1 + 5.01T + 7T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 - 5.01T + 13T^{2} \) |
| 17 | \( 1 - 1.91iT - 17T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 29 | \( 1 - 5.75T + 29T^{2} \) |
| 31 | \( 1 - 3.94iT - 31T^{2} \) |
| 37 | \( 1 + 8.21iT - 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 + 8.50T + 43T^{2} \) |
| 47 | \( 1 - 8.32iT - 47T^{2} \) |
| 53 | \( 1 + 1.72iT - 53T^{2} \) |
| 59 | \( 1 + 4.37iT - 59T^{2} \) |
| 61 | \( 1 - 2.56iT - 61T^{2} \) |
| 67 | \( 1 - 5.18T + 67T^{2} \) |
| 71 | \( 1 + 9.36iT - 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 - 0.143T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.558726116793946940893315694679, −7.58779793962462523900833153915, −6.78667293436861831671056601473, −6.20189136778860880393825139653, −5.67859961533678454804012933701, −4.66949850892345138902898056922, −3.61865158792468415986604390919, −3.25969116988903421490324367593, −2.53185012575067878256958695900, −0.872859958261247045505194072875,
0.44382198010491555241255378592, 1.35952311617897336149090489473, 2.83968957533906316110969030308, 3.17092718753105980530913173442, 4.11788616242991169363759699989, 5.31309954333077053627473178658, 5.81517637336530020665569506325, 6.61122009390095846004786933998, 7.06319851540983616760885117206, 7.959591889723640110653893677828
