| L(s) = 1 | + i·3-s + (−0.638 + 2.14i)5-s + 4.18i·7-s − 9-s − 1.89·11-s − i·13-s + (−2.14 − 0.638i)15-s + 1.73i·17-s − 1.11·19-s − 4.18·21-s − 9.30i·23-s + (−4.18 − 2.73i)25-s − i·27-s − 5.00·29-s − 10.0·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.285 + 0.958i)5-s + 1.58i·7-s − 0.333·9-s − 0.572·11-s − 0.277i·13-s + (−0.553 − 0.164i)15-s + 0.421i·17-s − 0.256·19-s − 0.912·21-s − 1.93i·23-s + (−0.836 − 0.547i)25-s − 0.192i·27-s − 0.929·29-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3574598401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3574598401\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.638 - 2.14i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 4.18iT - 7T^{2} \) |
| 11 | \( 1 + 1.89T + 11T^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 + 9.30iT - 23T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 8.02T + 41T^{2} \) |
| 43 | \( 1 + 1.00iT - 43T^{2} \) |
| 47 | \( 1 - 5.63iT - 47T^{2} \) |
| 53 | \( 1 - 0.174iT - 53T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 11.4iT - 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 + 4.01iT - 73T^{2} \) |
| 79 | \( 1 + 0.613T + 79T^{2} \) |
| 83 | \( 1 - 9.08iT - 83T^{2} \) |
| 89 | \( 1 - 1.46T + 89T^{2} \) |
| 97 | \( 1 + 3.85iT - 97T^{2} \) |
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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
−9.227996113104730985786996570670, −8.430040200833763333501722247364, −7.941479653635650076517886188295, −6.86046150471152003821110749141, −6.08880280131125451192939133631, −5.50652749977662708188419395614, −4.61129990543978068093860747397, −3.59819669982394422053920400338, −2.73773153272666257581810374645, −2.15944402601327588655241129860,
0.11570868621773449670516657342, 1.13558020667028500931193393394, 2.10605232538130170478785431461, 3.69231989406273872728974201795, 4.00850623803737120881697186291, 5.22790497976581137499229889599, 5.68367934815039810429792760299, 7.03131964693842700833849806030, 7.45920887933765210277257400966, 7.86657080866174027523381546900
