| L(s) = 1 | − i·3-s + 2·4-s + i·7-s + 2·9-s − 3·11-s − 2i·12-s − 5i·13-s + 4·16-s + 3i·17-s − 2·19-s + 21-s + 6i·23-s − 5i·27-s + 2i·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 4-s + 0.377i·7-s + 0.666·9-s − 0.904·11-s − 0.577i·12-s − 1.38i·13-s + 16-s + 0.727i·17-s − 0.458·19-s + 0.218·21-s + 1.25i·23-s − 0.962i·27-s + 0.377i·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35728 - 0.320411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35728 - 0.320411i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−12.86318801592704762278195545106, −11.71056418237357816472976395050, −10.69582023742138564708576503683, −9.898878392443386271295467110125, −8.147871851626506410832709692863, −7.54287668023440162427597917123, −6.36652082627462060432782413242, −5.32968870040299501719963145264, −3.26198202634456802309882085776, −1.81420037246371677867399420653,
2.15216858732523752878913131852, 3.84327066302090326621443783466, 5.12657998933874325446277157686, 6.69110116830137008686653654391, 7.37276997896353354604911032312, 8.815854534325350027084581673107, 10.08317058973632407671778685513, 10.69041874857904440775009233123, 11.67418408201091377603933954843, 12.67384449893208941937989550106
