L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.366 − 0.366i)7-s + (0.707 − 0.707i)8-s + 4.76i·11-s + (−3.46 − 3.46i)13-s − 0.517·14-s − 1.00·16-s + (1.03 + 1.03i)17-s − 7.46i·19-s + (3.36 − 3.36i)22-s + (1.41 − 1.41i)23-s + 4.89i·26-s + (0.366 + 0.366i)28-s + 6.31·29-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.138 − 0.138i)7-s + (0.250 − 0.250i)8-s + 1.43i·11-s + (−0.960 − 0.960i)13-s − 0.138·14-s − 0.250·16-s + (0.251 + 0.251i)17-s − 1.71i·19-s + (0.717 − 0.717i)22-s + (0.294 − 0.294i)23-s + 0.960i·26-s + (0.0691 + 0.0691i)28-s + 1.17·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.192728648\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.192728648\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.366 + 0.366i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.76iT - 11T^{2} \) |
| 13 | \( 1 + (3.46 + 3.46i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.03 - 1.03i)T + 17iT^{2} \) |
| 19 | \( 1 + 7.46iT - 19T^{2} \) |
| 23 | \( 1 + (-1.41 + 1.41i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.31T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 + (-2.19 + 2.19i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.62iT - 41T^{2} \) |
| 43 | \( 1 + (-1.26 - 1.26i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.24 - 4.24i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.08 - 5.08i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.52T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 + (-6.19 + 6.19i)T - 67iT^{2} \) |
| 71 | \( 1 + 15.4iT - 71T^{2} \) |
| 73 | \( 1 + (9.36 + 9.36i)T + 73iT^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + (1.46 - 1.46i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.93T + 89T^{2} \) |
| 97 | \( 1 + (-13.5 + 13.5i)T - 97iT^{2} \) |
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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
−9.566990172825196891555533342438, −8.836421059239605392845182194078, −7.78004806138461704733426174580, −7.30668339318953142741742562958, −6.36690853786085095723521594337, −4.92192719282493834171821220260, −4.49681970818806178366183599081, −2.99032685533505186678124107982, −2.25944493970122890474708629635, −0.73340809871335329737642347724,
1.03762372211956648975650616143, 2.45918251238146373361815162972, 3.69664873482844562861093778945, 4.85220997425000641415401527888, 5.73650444583437162278894032787, 6.48511343983482037276005277859, 7.33781160921093333590721991767, 8.348003028490573609227473312586, 8.615376727266055845708097088447, 9.927941001410014754800596664297