L(s) = 1 | + (−1 − i)3-s + (−1 − 2i)5-s + (−3 + 3i)7-s − i·9-s + 2i·11-s + (−3 + 3i)13-s + (−1 + 3i)15-s + (1 + i)17-s − 4·19-s + 6·21-s + (−1 − i)23-s + (−3 + 4i)25-s + (−4 + 4i)27-s − 10i·31-s + (2 − 2i)33-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.577i)3-s + (−0.447 − 0.894i)5-s + (−1.13 + 1.13i)7-s − 0.333i·9-s + 0.603i·11-s + (−0.832 + 0.832i)13-s + (−0.258 + 0.774i)15-s + (0.242 + 0.242i)17-s − 0.917·19-s + 1.30·21-s + (−0.208 − 0.208i)23-s + (−0.600 + 0.800i)25-s + (−0.769 + 0.769i)27-s − 1.79i·31-s + (0.348 − 0.348i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (1 + i)T + 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 + (-1 - i)T + 37iT^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + (5 + 5i)T + 43iT^{2} \) |
| 47 | \( 1 + (3 - 3i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5 + 5i)T - 53iT^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (-1 + i)T - 67iT^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (5 + 5i)T + 83iT^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−11.61035851327015145682374260297, −9.899112811866731620205860460292, −9.275419766142798099024364495762, −8.313855715264719559013996586430, −7.00377018629857202703587369608, −6.22634745121266893530599899908, −5.19330886075191238971953992096, −3.88278993611824687676963105714, −2.11840692599149381392305291938, 0,
2.97552235242153424367941095323, 3.92208554791867683175385982761, 5.20114434708407116346055847584, 6.46666421314551776853561233086, 7.22470515869465366961927750623, 8.281967966167249201909540105476, 9.946983819313847942203428222350, 10.31003028845863682024622747447, 10.98099863051018538828614691848