| L(s) = 1 | + 1.68·2-s + 1.73·3-s − 1.16·4-s − 5.25i·5-s + 2.91·6-s − 2.64i·7-s − 8.69·8-s + 2.99·9-s − 8.84i·10-s − 5.14i·11-s − 2.01·12-s − 16.5·13-s − 4.45i·14-s − 9.09i·15-s − 9.99·16-s + 5.08i·17-s + ⋯ |
| L(s) = 1 | + 0.842·2-s + 0.577·3-s − 0.290·4-s − 1.05i·5-s + 0.486·6-s − 0.377i·7-s − 1.08·8-s + 0.333·9-s − 0.884i·10-s − 0.468i·11-s − 0.167·12-s − 1.27·13-s − 0.318i·14-s − 0.606i·15-s − 0.624·16-s + 0.299i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.856448434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.856448434\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.73T \) |
| 7 | \( 1 + 2.64iT \) |
| 23 | \( 1 + (18.6 + 13.5i)T \) |
| good | 2 | \( 1 - 1.68T + 4T^{2} \) |
| 5 | \( 1 + 5.25iT - 25T^{2} \) |
| 11 | \( 1 + 5.14iT - 121T^{2} \) |
| 13 | \( 1 + 16.5T + 169T^{2} \) |
| 17 | \( 1 - 5.08iT - 289T^{2} \) |
| 19 | \( 1 + 18.9iT - 361T^{2} \) |
| 29 | \( 1 + 25.1T + 841T^{2} \) |
| 31 | \( 1 - 30.8T + 961T^{2} \) |
| 37 | \( 1 - 5.29iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.11T + 1.68e3T^{2} \) |
| 43 | \( 1 + 66.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 1.22T + 2.20e3T^{2} \) |
| 53 | \( 1 - 4.56iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 48.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 23.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 54.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 74.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 75.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 146. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 37.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 91.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 183. iT - 9.40e3T^{2} \) |
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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
−10.33239640186157860942310856968, −9.385117441457808339822823753073, −8.743027031855754730102941719094, −7.87638285394914910545173722927, −6.64190870848032926907658017533, −5.34956505664368543820239609846, −4.64975329591251378001770735501, −3.77777871844163732021730611691, −2.45530357151988599372070928818, −0.51476468313289834405430375399,
2.29086291906648498950803742722, 3.19719681611104358532363422964, 4.23714501221885323300576721970, 5.31178518070725926091281728215, 6.36876900975800700119525028090, 7.38279103597843786702995902059, 8.277877567449241229671697226622, 9.623775880536245458319152397427, 9.880086254667067896764326780063, 11.26009427035195281064432699291
