L(s) = 1 | + 3-s + 9-s + (1 + i)11-s + 2i·17-s + (−1 − i)19-s − i·25-s + 27-s + (1 + i)33-s + (−1 + i)43-s + 49-s + 2i·51-s + (−1 − i)57-s + (−1 − i)59-s + (−1 − i)67-s − i·75-s + ⋯ |
L(s) = 1 | + 3-s + 9-s + (1 + i)11-s + 2i·17-s + (−1 − i)19-s − i·25-s + 27-s + (1 + i)33-s + (−1 + i)43-s + 49-s + 2i·51-s + (−1 − i)57-s + (−1 − i)59-s + (−1 − i)67-s − i·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.883387391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.883387391\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
good | 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-1 - i)T + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (1 + i)T + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-1 + i)T - iT^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.889169388916295252295546034950, −8.278518913699956095155232089547, −7.57835301984091526885560966224, −6.59974280053884547985669621562, −6.27379427249793469489236316305, −4.71392230783152933257892793228, −4.22967984828198272831790078940, −3.42225074320931237966468349963, −2.25214638278143473372484537808, −1.56971698592961988803804701966,
1.19374793950770575579742669565, 2.34616180650914776357796903068, 3.29686819845482800439453457706, 3.90887775968505128405381294931, 4.87392893941052359135438913999, 5.85858686958741106402975018290, 6.76716017620525827439057530610, 7.38439463536526416493683727925, 8.187908413811028477961657965379, 8.991867241591200503274452823422