L(s) = 1 | + (−0.382 + 0.923i)3-s − 4-s + 1.84·5-s + (−0.707 − 0.707i)9-s + i·11-s + (0.382 − 0.923i)12-s + (−0.707 + 1.70i)15-s + 16-s − 1.84·20-s + 1.41i·23-s + 2.41·25-s + (0.923 − 0.382i)27-s − 0.765i·31-s + (−0.923 − 0.382i)33-s + (0.707 + 0.707i)36-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)3-s − 4-s + 1.84·5-s + (−0.707 − 0.707i)9-s + i·11-s + (0.382 − 0.923i)12-s + (−0.707 + 1.70i)15-s + 16-s − 1.84·20-s + 1.41i·23-s + 2.41·25-s + (0.923 − 0.382i)27-s − 0.765i·31-s + (−0.923 − 0.382i)33-s + (0.707 + 0.707i)36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.046549088\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.046549088\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - iT \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 - 1.84T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 0.765iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 0.765T + T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.765T + T^{2} \) |
| 97 | \( 1 + 1.84iT - T^{2} \) |
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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
−9.771170037108966007770384612620, −9.303812252300225797753710115639, −8.582351810475581580337852752789, −7.29967814098500318214972648886, −6.13553805900387566812368783370, −5.57034218105287184214200374641, −4.92113006035103134806694209061, −4.10683777032202789034677942801, −2.88828605168723198900361559424, −1.54785031717955117539781256490,
0.983409077516889507731898657844, 2.13347694220110486998580563407, 3.20184263750341287248936640681, 4.81952138589266903924196226809, 5.41805485894364728721709838285, 6.18069387153598171444242559877, 6.68562381780977312985875891172, 8.007415759038859089344953418159, 8.720230768368919122572537044813, 9.260578343396553424189560883498