| L(s) = 1 | + i·3-s + 1.19i·5-s − 0.613·7-s − 9-s − 1.53i·11-s − 1.24i·13-s − 1.19·15-s − 4.35·17-s − 1.29i·19-s − 0.613i·21-s + 4·23-s + 3.56·25-s − i·27-s − 0.965i·29-s − 6.77·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.536i·5-s − 0.231·7-s − 0.333·9-s − 0.461i·11-s − 0.346i·13-s − 0.309·15-s − 1.05·17-s − 0.297i·19-s − 0.133i·21-s + 0.834·23-s + 0.712·25-s − 0.192i·27-s − 0.179i·29-s − 1.21·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6719154521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6719154521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 - 1.19iT - 5T^{2} \) |
| 7 | \( 1 + 0.613T + 7T^{2} \) |
| 11 | \( 1 + 1.53iT - 11T^{2} \) |
| 13 | \( 1 + 1.24iT - 13T^{2} \) |
| 17 | \( 1 + 4.35T + 17T^{2} \) |
| 19 | \( 1 + 1.29iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 0.965iT - 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 + 8.68iT - 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 + 9.04iT - 59T^{2} \) |
| 61 | \( 1 + 1.52iT - 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 3.56T + 71T^{2} \) |
| 73 | \( 1 + 5.15T + 73T^{2} \) |
| 79 | \( 1 - 7.49T + 79T^{2} \) |
| 83 | \( 1 + 7.18iT - 83T^{2} \) |
| 89 | \( 1 - 0.672T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−8.623795242415021541874465313368, −7.88133010348825059419713688287, −6.64993526420409926208057813822, −6.58920650276755975228644348519, −5.23364717259012212300617290837, −4.80118966434060839233119059041, −3.51200422178362141922569878470, −3.12304240290548064519677151567, −1.91603407508617392232168752606, −0.20994494818707011428242107402,
1.25083482346848272024494521700, 2.18190007318647126490116933323, 3.23715443071158827920329535843, 4.31921607103525214695826662980, 5.03245012887909565138067548603, 5.91587382163539251242677951109, 6.79349611378544533428297727418, 7.25697922951923769929383606885, 8.175426592660023929802366802369, 8.979775400618680655273226072835
