| L(s) = 1 | − i·3-s + 2i·7-s − 9-s − 4·11-s + i·13-s − 2i·17-s + 2·19-s + 2·21-s + i·27-s + 6·29-s − 10·31-s + 4i·33-s − 10i·37-s + 39-s + 8·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.755i·7-s − 0.333·9-s − 1.20·11-s + 0.277i·13-s − 0.485i·17-s + 0.458·19-s + 0.436·21-s + 0.192i·27-s + 1.11·29-s − 1.79·31-s + 0.696i·33-s − 1.64i·37-s + 0.160·39-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.415150847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.415150847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.198937019865660966338759335951, −7.64230268158386858570712331887, −6.99033619779418207256481860607, −6.03737650264777878047540587594, −5.44481559349346003748052322742, −4.76078851271347277465880954350, −3.53476094087088779480373907759, −2.61347705753606061775168101310, −1.97610104160655761645026826401, −0.51504408080624484691603387277,
0.881909319301179468339753103370, 2.30954123229753188494097466945, 3.23688193190466955137982623707, 3.99279750073826273698741766308, 4.87545886170949793420720091993, 5.46959890701307364531605799044, 6.33704629284539194715826325860, 7.29960857867754195702741577097, 7.84485410849878976438568082881, 8.569060842297272741035026183049
