L(s) = 1 | − i·2-s + 3.09i·3-s − 4-s + 0.646i·5-s + 3.09·6-s + (−2.44 + i)7-s + i·8-s − 6.58·9-s + 0.646·10-s + (2.79 + 1.79i)11-s − 3.09i·12-s + 3.09·13-s + (1 + 2.44i)14-s − 2·15-s + 16-s − 3.74·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.78i·3-s − 0.5·4-s + 0.288i·5-s + 1.26·6-s + (−0.925 + 0.377i)7-s + 0.353i·8-s − 2.19·9-s + 0.204·10-s + (0.841 + 0.540i)11-s − 0.893i·12-s + 0.858·13-s + (0.267 + 0.654i)14-s − 0.516·15-s + 0.250·16-s − 0.907·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.743638 + 0.618671i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.743638 + 0.618671i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 7 | \( 1 + (2.44 - i)T \) |
| 11 | \( 1 + (-2.79 - 1.79i)T \) |
good | 3 | \( 1 - 3.09iT - 3T^{2} \) |
| 5 | \( 1 - 0.646iT - 5T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.58iT - 29T^{2} \) |
| 31 | \( 1 - 1.15iT - 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 5.03iT - 47T^{2} \) |
| 53 | \( 1 + 2.41T + 53T^{2} \) |
| 59 | \( 1 + 3.09iT - 59T^{2} \) |
| 61 | \( 1 - 9.28T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 6.32T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 9.15T + 83T^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 97 | \( 1 - 15.9iT - 97T^{2} \) |
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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
−13.16980065844479182833777197970, −11.79299309669127043446451957901, −11.08586651728769404271460757128, −10.06362438232702606717641000162, −9.412583555625762001907199884216, −8.686325028583903411631354757140, −6.52142185537106594181065793047, −5.12777921881293340631834055990, −3.92954665589412620179875441003, −3.01030818027584223214623570382,
1.05786638568198291507047137292, 3.35305905845959677044757790650, 5.52658940533424529287596662615, 6.73122133288022275870403371695, 7.01588213496961311123316421908, 8.474411334108633125044012128544, 9.122775664915696703040962565952, 10.96433173091296011038007037030, 12.12391859048447971869636795696, 12.98535177570740870978962251914