| L(s) = 1 | + i·3-s + 2i·5-s + 4.46·7-s + 2·9-s + 4i·11-s + 4.46i·13-s − 2·15-s − 7.92·17-s + i·19-s + 4.46i·21-s − 6.46·23-s + 25-s + 5i·27-s − 2.46i·29-s − 4.92·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.894i·5-s + 1.68·7-s + 0.666·9-s + 1.20i·11-s + 1.23i·13-s − 0.516·15-s − 1.92·17-s + 0.229i·19-s + 0.974i·21-s − 1.34·23-s + 0.200·25-s + 0.962i·27-s − 0.457i·29-s − 0.885·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.004121574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004121574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 4.46iT - 13T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 23 | \( 1 + 6.46T + 23T^{2} \) |
| 29 | \( 1 + 2.46iT - 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 4.92iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 12.4iT - 53T^{2} \) |
| 59 | \( 1 - 9.92iT - 59T^{2} \) |
| 61 | \( 1 + 8.92iT - 61T^{2} \) |
| 67 | \( 1 + 5.92iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 2.92iT - 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−9.250913267873731324159148800647, −8.563854221957067457893530951096, −7.49768237286940300650729955832, −7.10266985328086863558981497371, −6.25970916858503109882065644036, −4.98470551760517227454407708024, −4.37842987921204996024211464808, −3.96293995075191885875870140489, −2.13543319207443326434944819408, −1.89945010595042956919903016402,
0.66941695009855132386542689463, 1.56296741016349353976830580661, 2.52670080657890551209349845038, 4.07151891120267244705872548144, 4.65330180057236352583075441693, 5.51027057401408236049141436170, 6.22147316727245339399194186664, 7.44430276496043768340318782945, 7.85452968457555741286376312264, 8.701017006304712672465124415552
