L(s) = 1 | + 15.5·3-s − 196i·5-s + 270. i·7-s + 243·9-s + 1.72e3·11-s − 2.88e3i·13-s − 3.05e3i·15-s − 2.89e3·17-s + 9.78e3·19-s + 4.21e3i·21-s + 1.62e3i·23-s − 2.27e4·25-s + 3.78e3·27-s − 1.45e4i·29-s + 6.87e3i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.56i·5-s + 0.787i·7-s + 0.333·9-s + 1.29·11-s − 1.31i·13-s − 0.905i·15-s − 0.589·17-s + 1.42·19-s + 0.454i·21-s + 0.133i·23-s − 1.45·25-s + 0.192·27-s − 0.598i·29-s + 0.230i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.967833748\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.967833748\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 15.5T \) |
good | 5 | \( 1 + 196iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 270. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.72e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 2.88e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.89e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 9.78e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.62e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.45e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 6.87e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.21e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.87e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.35e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.51e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.44e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 3.33e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.33e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 3.19e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 4.20e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.28e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.91e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 3.57e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.37e6T + 4.96e11T^{2} \) |
| 97 | \( 1 - 4.89e5T + 8.32e11T^{2} \) |
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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
−9.588045234722053989459598115039, −9.232563188775874291836913267189, −8.398422464593708796940829041730, −7.60701801560782825167327054895, −6.09200837984116699002760829127, −5.20248663862540180314850241707, −4.21204837772251925100569426271, −2.98500428587598290081915424481, −1.57187398046978677362870380447, −0.66696729893060139318968681303,
1.24665184994561214639361025379, 2.48133115201043464525156429490, 3.58851203622710696686237322064, 4.28314039892719423353218960884, 6.12679983564161528881908034032, 7.09043334192746213042732190944, 7.32567985154722438331129218697, 8.916227655012931344384934634969, 9.606823343126129623588209043220, 10.59951580608954767289989381590