L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − i·7-s − i·8-s − 9-s + i·12-s − 2i·13-s + 14-s + 16-s + 3i·17-s − i·18-s − 2·19-s − 21-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.377i·7-s − 0.353i·8-s − 0.333·9-s + 0.288i·12-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s + 0.727i·17-s − 0.235i·18-s − 0.458·19-s − 0.218·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9096997840\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9096997840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 11iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.605051765250741409100546314634, −7.962259243959860198193963418122, −7.34147782862468235489264738559, −6.71776481873990189119981420432, −5.83232504593998188376952984720, −5.36206780451406251723517096260, −4.19488962960997314416677710871, −3.50989211916932776681356517836, −2.25458655580470329699796194227, −1.09368263445562024420625367736,
0.29830048807557271985178934611, 1.91755864360957928797763209637, 2.61188766267494037976637182749, 3.82065648193455679506069218430, 4.16364353677489225796839222990, 5.39800810465250056963474902579, 5.68629807227761166986993999394, 6.98347171447754643438213838256, 7.64932742460707046542319130297, 8.718237525568172304344904483514