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 | \( 1 - 14.2iT - 97T^{2} \) |
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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
−7.959591889723640110653893677828, −7.06319851540983616760885117206, −6.61122009390095846004786933998, −5.81517637336530020665569506325, −5.31309954333077053627473178658, −4.11788616242991169363759699989, −3.17092718753105980530913173442, −2.83968957533906316110969030308, −1.35952311617897336149090489473, −0.44382198010491555241255378592,
0.872859958261247045505194072875, 2.53185012575067878256958695900, 3.25969116988903421490324367593, 3.61865158792468415986604390919, 4.66949850892345138902898056922, 5.67859961533678454804012933701, 6.20189136778860880393825139653, 6.78667293436861831671056601473, 7.58779793962462523900833153915, 8.558726116793946940893315694679