| L(s) = 1 | + 1.73i·3-s + 1.08·5-s − 2.64i·7-s − 2.99·9-s − 3.75i·11-s − 10.2·13-s + 1.88i·15-s − 19.5·17-s − 9.33i·19-s + 4.58·21-s − 1.06i·23-s − 23.8·25-s − 5.19i·27-s − 0.156·29-s + 15.1i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.217·5-s − 0.377i·7-s − 0.333·9-s − 0.341i·11-s − 0.787·13-s + 0.125i·15-s − 1.15·17-s − 0.491i·19-s + 0.218·21-s − 0.0462i·23-s − 0.952·25-s − 0.192i·27-s − 0.00540·29-s + 0.487i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3895063779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3895063779\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 - 1.08T + 25T^{2} \) |
| 11 | \( 1 + 3.75iT - 121T^{2} \) |
| 13 | \( 1 + 10.2T + 169T^{2} \) |
| 17 | \( 1 + 19.5T + 289T^{2} \) |
| 19 | \( 1 + 9.33iT - 361T^{2} \) |
| 23 | \( 1 + 1.06iT - 529T^{2} \) |
| 29 | \( 1 + 0.156T + 841T^{2} \) |
| 31 | \( 1 - 15.1iT - 961T^{2} \) |
| 37 | \( 1 + 20.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 43.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 21.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 21.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 30.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 59.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 58.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 84.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 12.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 106. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 128. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 100.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 101.T + 9.40e3T^{2} \) |
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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
−10.01131790670100665334506842938, −9.190189255623221359570377629380, −8.397992631254137101522585787946, −7.28613672527585826505093666292, −6.41538130825814747978857733884, −5.27176386577211730897443187749, −4.44928951137695548684045828961, −3.34778508429709977583931786418, −2.06818242760956466677166614096, −0.12871883228029944496751002013,
1.73865454077464573299281120277, 2.68731140514758713753568139053, 4.17255738505170895016498164663, 5.29192175731649400237073299840, 6.22960163342259261590181827156, 7.10699695096288397269573672522, 7.954869099985049841923537714136, 8.896699141819923601516681544670, 9.710078996135071020645230246254, 10.59696412099043406696139606946
