L(s) = 1 | + (−1 + 1.41i)3-s + 2i·5-s + (−1.00 − 2.82i)9-s + 4.24·11-s + 1.41·13-s + (−2.82 − 2i)15-s − 2i·17-s + 4.24·23-s + 25-s + (5.00 + 1.41i)27-s − 8.48i·29-s + 8.48i·31-s + (−4.24 + 6i)33-s + 6·37-s + (−1.41 + 2.00i)39-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + 0.894i·5-s + (−0.333 − 0.942i)9-s + 1.27·11-s + 0.392·13-s + (−0.730 − 0.516i)15-s − 0.485i·17-s + 0.884·23-s + 0.200·25-s + (0.962 + 0.272i)27-s − 1.57i·29-s + 1.52i·31-s + (−0.738 + 1.04i)33-s + 0.986·37-s + (−0.226 + 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.646624302\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646624302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−9.028450098903742896631820436767, −8.709113353154324499557436837101, −7.22279719972251926137315928341, −6.78523287089224086867914167926, −5.98670371741266532482946664618, −5.19544374056313495832522896435, −4.13113444317770751652418728048, −3.56986651378450047748906429905, −2.52628119346454028785751727899, −0.886425458672409998414548296656,
0.947312897402252552708280492524, 1.54770447088838743360439621952, 2.96179013381296062116482403623, 4.24308495911181245228612306380, 4.89955661027194404829139866549, 5.93303045254508675153672203626, 6.40690404072027257896412183166, 7.30269619352726912958258330147, 8.080047222573335620674945009730, 8.869219173026740937370496608845