| L(s) = 1 | + 1.54i·5-s + 2.64·7-s − 5.28i·11-s + 17.4·13-s + 7.16i·17-s − 12.1·19-s − 41.2i·23-s + 22.6·25-s − 38.3i·29-s − 45.2·31-s + 4.08i·35-s − 52.5·37-s − 64.3i·41-s − 40.6·43-s + 79.0i·47-s + ⋯ |
| L(s) = 1 | + 0.308i·5-s + 0.377·7-s − 0.480i·11-s + 1.34·13-s + 0.421i·17-s − 0.639·19-s − 1.79i·23-s + 0.904·25-s − 1.32i·29-s − 1.45·31-s + 0.116i·35-s − 1.42·37-s − 1.56i·41-s − 0.946·43-s + 1.68i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.169316717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169316717\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 1.54iT - 25T^{2} \) |
| 11 | \( 1 + 5.28iT - 121T^{2} \) |
| 13 | \( 1 - 17.4T + 169T^{2} \) |
| 17 | \( 1 - 7.16iT - 289T^{2} \) |
| 19 | \( 1 + 12.1T + 361T^{2} \) |
| 23 | \( 1 + 41.2iT - 529T^{2} \) |
| 29 | \( 1 + 38.3iT - 841T^{2} \) |
| 31 | \( 1 + 45.2T + 961T^{2} \) |
| 37 | \( 1 + 52.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 64.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 79.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 88.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 39.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 13.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 62.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 114.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 42.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 9.41iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 63.2T + 9.40e3T^{2} \) |
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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
−8.170478142617303399323719392971, −7.28463481969906771620627124539, −6.36953626339317683248854965434, −6.03036980493052568764765685578, −4.98678771537883417268744721470, −4.13090781184627148838635272855, −3.42431404741993393746603519436, −2.40837662691366466455524589791, −1.43400772010840695438132652158, −0.23840363732616093626411654439,
1.27329834252826190550639459159, 1.86518711171746635062948535450, 3.30004263098252592166996074108, 3.80403163222020821221769032686, 5.02818744052679946331499713216, 5.29961952871187575360621381679, 6.41807527084431460896235381252, 7.05124028168302407610847891574, 7.81265372472077609139306683180, 8.704171417633046642133279212552
