L(s) = 1 | + (−1.70 + 0.292i)3-s + (−2.12 − 0.707i)5-s + (3 + 3i)7-s + (2.82 − i)9-s − 2.82i·11-s + (−3 + 3i)13-s + (3.82 + 0.585i)15-s + (1.41 − 1.41i)17-s − 4i·19-s + (−5.99 − 4.24i)21-s + (0.707 + 0.707i)23-s + (3.99 + 3i)25-s + (−4.53 + 2.53i)27-s + 2.82·29-s − 4·31-s + ⋯ |
L(s) = 1 | + (−0.985 + 0.169i)3-s + (−0.948 − 0.316i)5-s + (1.13 + 1.13i)7-s + (0.942 − 0.333i)9-s − 0.852i·11-s + (−0.832 + 0.832i)13-s + (0.988 + 0.151i)15-s + (0.342 − 0.342i)17-s − 0.917i·19-s + (−1.30 − 0.925i)21-s + (0.147 + 0.147i)23-s + (0.799 + 0.600i)25-s + (−0.872 + 0.487i)27-s + 0.525·29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9359825396\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9359825396\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.292i)T \) |
| 5 | \( 1 + (2.12 + 0.707i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-4 - 4i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 + (-5 + 5i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.65 - 5.65i)T - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-7 - 7i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.5iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 + (-12 - 12i)T + 97iT^{2} \) |
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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
−9.611673972765655990661254492423, −8.930541496562653996250964380711, −8.126306321415690249518052354023, −7.31553042858734730465767504215, −6.39903365804757882104017711584, −5.25089025812466061772392331390, −4.94284588916573329290900683996, −3.97141231131997316245220540232, −2.54746560589823977563982490788, −1.04513868396122868777875087781,
0.55382754662239501662072083754, 1.85705099346706542136574653442, 3.58002301685621773609168675374, 4.50571862223244721734267275288, 5.00082338016882856162623672124, 6.17283665100780880612427368737, 7.36386614598318389887942367354, 7.48405091783275484492084675321, 8.232103659627459761501589847723, 9.768266581293874940970511323998