| L(s) = 1 | + 0.460i·3-s + 1.55i·5-s − 7-s + 2.78·9-s + 5.69i·11-s − 0.801i·13-s − 0.717·15-s + 4.85·17-s + 4.04i·19-s − 0.460i·21-s + 5.34·23-s + 2.57·25-s + 2.66i·27-s − 8.47i·29-s + 10.9·31-s + ⋯ |
| L(s) = 1 | + 0.265i·3-s + 0.696i·5-s − 0.377·7-s + 0.929·9-s + 1.71i·11-s − 0.222i·13-s − 0.185·15-s + 1.17·17-s + 0.928i·19-s − 0.100i·21-s + 1.11·23-s + 0.514·25-s + 0.512i·27-s − 1.57i·29-s + 1.95·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.096664480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.096664480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 0.460iT - 3T^{2} \) |
| 5 | \( 1 - 1.55iT - 5T^{2} \) |
| 11 | \( 1 - 5.69iT - 11T^{2} \) |
| 13 | \( 1 + 0.801iT - 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 - 4.04iT - 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 29 | \( 1 + 8.47iT - 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 7.26iT - 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 6.90iT - 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 - 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 1.61iT - 59T^{2} \) |
| 61 | \( 1 - 7.21iT - 61T^{2} \) |
| 67 | \( 1 - 9.44iT - 67T^{2} \) |
| 71 | \( 1 - 1.51T + 71T^{2} \) |
| 73 | \( 1 + 4.21T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 - 6.11iT - 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.801533706059054765523712614650, −7.76865893871628489980149952622, −7.24497056923216482117410466388, −6.69134390606954218753927111262, −5.77486905024185867425942732164, −4.81379390052177748609278425241, −4.15201817913463915598832004616, −3.25898652784446658138782004953, −2.34099395200389130768480300166, −1.21845107419357675305872318148,
0.75516243458689178242861348379, 1.39086809417698373217890997726, 3.03398801264533198881739996245, 3.42795488189252571333636931777, 4.92144479528885527323961502539, 5.01864833608108778659994906982, 6.44374498075024587365921547558, 6.63158570396912925091279179253, 7.73314977438863119915298041371, 8.472292748520608010489950119699
