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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8255054004\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8255054004\) |
\(L(1)\) |
|
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
−8.803168269050532159351772513753, −8.126544118231910473478541616356, −7.47583931653308452568269025400, −6.47537485738457200164339407308, −5.73292479242823631303042864156, −5.45892221723750339750860124333, −4.22964232404430460503467035321, −3.57151940899356055238328739902, −2.25057310954275461927680389280, −0.988624387561853365470562512342,
0.75825816710647015445463612734, 2.21222672955083138994476016064, 3.17825728774416267984895776629, 4.52430033146478063012974348020, 5.05058586032667330641143338617, 5.56503528388875144518949146147, 6.72810762523987802388296506861, 7.32920848998475293440584309054, 7.69989132846954318099785318847, 9.134789036095722181706723046940