| L(s) = 1 | + i·2-s − 4-s − 2.23i·5-s + 3.23i·7-s − i·8-s + 2.23·10-s − 4.47·11-s − i·13-s − 3.23·14-s + 16-s − 7.23i·17-s + 2.76·19-s + 2.23i·20-s − 4.47i·22-s + 2.76i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.999i·5-s + 1.22i·7-s − 0.353i·8-s + 0.707·10-s − 1.34·11-s − 0.277i·13-s − 0.864·14-s + 0.250·16-s − 1.75i·17-s + 0.634·19-s + 0.499i·20-s − 0.953i·22-s + 0.576i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6234787472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6234787472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 3.23iT - 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 + 7.23iT - 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 - 2.76iT - 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10.9iT - 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 - 2.47iT - 43T^{2} \) |
| 47 | \( 1 + 12.9iT - 47T^{2} \) |
| 53 | \( 1 + 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 13.2iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 4.94iT - 83T^{2} \) |
| 89 | \( 1 - 0.472T + 89T^{2} \) |
| 97 | \( 1 + 3.70iT - 97T^{2} \) |
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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
−9.174601610690181129854815098713, −8.933924204698048386180301046018, −7.77634618559012901780365375497, −7.40085379561879823170767833345, −5.90130607404530129969061503469, −5.29936802720990945089762401949, −4.89888016355259695651438855418, −3.35219085642063347919301023193, −2.14464046230874063237771012138, −0.26201167215684390288338673534,
1.57933089937505286183862185698, 2.84194299343676217612214822296, 3.67555559451482835302121707747, 4.56119144426292483518143247032, 5.76403880007138455769491397093, 6.72184024052943561211342506970, 7.66098698629732074408746764982, 8.175923495633132699666901056009, 9.491071738116274701060013647397, 10.33175685116791975373319095871
