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
−10.98099863051018538828614691848, −10.31003028845863682024622747447, −9.946983819313847942203428222350, −8.281967966167249201909540105476, −7.22470515869465366961927750623, −6.46666421314551776853561233086, −5.20114434708407116346055847584, −3.92208554791867683175385982761, −2.97552235242153424367941095323, 0,
2.11840692599149381392305291938, 3.88278993611824687676963105714, 5.19330886075191238971953992096, 6.22634745121266893530599899908, 7.00377018629857202703587369608, 8.313855715264719559013996586430, 9.275419766142798099024364495762, 9.899112811866731620205860460292, 11.61035851327015145682374260297