| L(s) = 1 | + (0.869 + 1.49i)3-s + 5-s + (0.807 − 2.51i)7-s + (−1.48 + 2.60i)9-s + 0.541i·11-s − 5.85i·13-s + (0.869 + 1.49i)15-s + 6.22·17-s + 5.74i·19-s + (4.47 − 0.978i)21-s − 6.55i·23-s + 25-s + (−5.19 + 0.0312i)27-s − 5.28i·29-s + 8.99i·31-s + ⋯ |
| L(s) = 1 | + (0.501 + 0.865i)3-s + 0.447·5-s + (0.305 − 0.952i)7-s + (−0.496 + 0.868i)9-s + 0.163i·11-s − 1.62i·13-s + (0.224 + 0.386i)15-s + 1.50·17-s + 1.31i·19-s + (0.976 − 0.213i)21-s − 1.36i·23-s + 0.200·25-s + (−0.999 + 0.00601i)27-s − 0.981i·29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.379947026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.379947026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.869 - 1.49i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-0.807 + 2.51i)T \) |
| good | 11 | \( 1 - 0.541iT - 11T^{2} \) |
| 13 | \( 1 + 5.85iT - 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 - 5.74iT - 19T^{2} \) |
| 23 | \( 1 + 6.55iT - 23T^{2} \) |
| 29 | \( 1 + 5.28iT - 29T^{2} \) |
| 31 | \( 1 - 8.99iT - 31T^{2} \) |
| 37 | \( 1 - 7.33T + 37T^{2} \) |
| 41 | \( 1 - 5.16T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 + 9.75iT - 53T^{2} \) |
| 59 | \( 1 - 9.76T + 59T^{2} \) |
| 61 | \( 1 - 0.433iT - 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 - 5.25iT - 71T^{2} \) |
| 73 | \( 1 - 9.42iT - 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 - 3.24iT - 97T^{2} \) |
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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
−9.709235669489120200488216858030, −8.296670819163837988590986892487, −8.134573405893498781348066918432, −7.18080226242334213010907419984, −5.87479474228494611498528785737, −5.29105838264546251899438531566, −4.27611543874641319502570219224, −3.47553934182574777562658948048, −2.56461827454425141416280580587, −1.02965986978929773574792976989,
1.24497840310705636321428074131, 2.21160277127196219098098652548, 3.03187328667130520728117687977, 4.27694159599161398500437674821, 5.52564964359300309924431961097, 6.04710725912963073383435601582, 7.05905703919273937570675007879, 7.66900744001224272865972146770, 8.608935777280003908670464518052, 9.354096651584025574634554131489
