L(s) = 1 | + (1.73 − 0.0835i)3-s + (0.431 + 0.431i)5-s + 3.10·7-s + (2.98 − 0.289i)9-s + (−2.98 + 2.98i)11-s + (−2.10 − 2.10i)13-s + (0.782 + 0.710i)15-s − 2.42i·17-s + (−0.710 + 0.710i)19-s + (5.36 − 0.259i)21-s + 5.97i·23-s − 4.62i·25-s + (5.14 − 0.749i)27-s + (2.86 − 2.86i)29-s − 0.524i·31-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0482i)3-s + (0.193 + 0.193i)5-s + 1.17·7-s + (0.995 − 0.0963i)9-s + (−0.900 + 0.900i)11-s + (−0.583 − 0.583i)13-s + (0.202 + 0.183i)15-s − 0.589i·17-s + (−0.163 + 0.163i)19-s + (1.17 − 0.0565i)21-s + 1.24i·23-s − 0.925i·25-s + (0.989 − 0.144i)27-s + (0.531 − 0.531i)29-s − 0.0941i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03427 + 0.101265i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03427 + 0.101265i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.73 + 0.0835i)T \) |
good | 5 | \( 1 + (-0.431 - 0.431i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + (2.98 - 2.98i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.10 + 2.10i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.42iT - 17T^{2} \) |
| 19 | \( 1 + (0.710 - 0.710i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.97iT - 23T^{2} \) |
| 29 | \( 1 + (-2.86 + 2.86i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.524iT - 31T^{2} \) |
| 37 | \( 1 + (1.52 - 1.52i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.81T + 41T^{2} \) |
| 43 | \( 1 + (-0.710 - 0.710i)T + 43iT^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 + (8.83 + 8.83i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.0804 + 0.0804i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5.72 - 5.72i)T + 61iT^{2} \) |
| 67 | \( 1 + (0.391 - 0.391i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.01iT - 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 3.47iT - 79T^{2} \) |
| 83 | \( 1 + (4.55 + 4.55i)T + 83iT^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.67T + 97T^{2} \) |
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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
−11.33151473859738608671178359354, −10.15800415081975468629441840462, −9.695200953019127861347370240252, −8.316909139825110152538144070345, −7.82733104583920713149585938601, −6.94685753810452472492881099803, −5.27917337699648326273601976989, −4.44307221071707749861483037887, −2.88919319469775757278834614569, −1.84905941962383674710749493756,
1.70922024121752518867729764434, 2.94112867655404584736742880958, 4.38141852817124600300172831029, 5.26289972876826973602011173835, 6.76058147555836096008427807350, 7.931505677831497623990404751997, 8.424916099933025715898516199647, 9.316245005462753058007596422632, 10.45875912377075377329570030692, 11.12723543390617868066882109345