| L(s) = 1 | + (1.72 + 0.0964i)3-s − 5-s + (−0.208 − 2.63i)7-s + (2.98 + 0.333i)9-s − 1.71i·11-s − 3.11i·13-s + (−1.72 − 0.0964i)15-s − 3.59·17-s + 1.04i·19-s + (−0.106 − 4.58i)21-s + 0.587i·23-s + 25-s + (5.12 + 0.864i)27-s − 4.47i·29-s − 8.51i·31-s + ⋯ |
| L(s) = 1 | + (0.998 + 0.0556i)3-s − 0.447·5-s + (−0.0789 − 0.996i)7-s + (0.993 + 0.111i)9-s − 0.516i·11-s − 0.863i·13-s + (−0.446 − 0.0249i)15-s − 0.871·17-s + 0.239i·19-s + (−0.0232 − 0.999i)21-s + 0.122i·23-s + 0.200·25-s + (0.986 + 0.166i)27-s − 0.830i·29-s − 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0232 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0232 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.923162031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923162031\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 - 0.0964i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (0.208 + 2.63i)T \) |
| good | 11 | \( 1 + 1.71iT - 11T^{2} \) |
| 13 | \( 1 + 3.11iT - 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 - 1.04iT - 19T^{2} \) |
| 23 | \( 1 - 0.587iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 8.51iT - 31T^{2} \) |
| 37 | \( 1 + 7.99T + 37T^{2} \) |
| 41 | \( 1 + 7.74T + 41T^{2} \) |
| 43 | \( 1 - 5.05T + 43T^{2} \) |
| 47 | \( 1 - 9.90T + 47T^{2} \) |
| 53 | \( 1 - 4.63iT - 53T^{2} \) |
| 59 | \( 1 + 2.11T + 59T^{2} \) |
| 61 | \( 1 + 8.11iT - 61T^{2} \) |
| 67 | \( 1 - 8.80T + 67T^{2} \) |
| 71 | \( 1 + 2.57iT - 71T^{2} \) |
| 73 | \( 1 + 6.88iT - 73T^{2} \) |
| 79 | \( 1 - 7.01T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 - 8.17T + 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−9.073275325722485659680126670920, −8.266572659384043436649495395214, −7.69695341556660331269070093744, −7.02655957660536113723568168136, −6.02516755059277447818438463307, −4.75513103825298602240582735508, −3.92302307887008839746415421048, −3.29491900712710228201738191165, −2.13354728065075589341596798556, −0.63328058229575895244439884079,
1.68417107740999926222032339886, 2.55173181389408608839186003434, 3.53656423299114178950472865736, 4.47681334028520192454443901557, 5.30248834713268339026056946381, 6.76761889851034536360165233207, 7.00345924625538147673551529792, 8.192294833469220962358753230388, 8.857603123502445270584139673488, 9.174430150089547070580607083918
