L(s) = 1 | + 3.23i·3-s + (−1 + 2i)5-s + 4.47i·7-s − 7.47·9-s − 1.23i·13-s + (−6.47 − 3.23i)15-s + 2.47i·17-s + 19-s − 14.4·21-s + 2i·23-s + (−3 − 4i)25-s − 14.4i·27-s − 2·29-s + 10.4·31-s + (−8.94 − 4.47i)35-s + ⋯ |
L(s) = 1 | + 1.86i·3-s + (−0.447 + 0.894i)5-s + 1.69i·7-s − 2.49·9-s − 0.342i·13-s + (−1.67 − 0.835i)15-s + 0.599i·17-s + 0.229·19-s − 3.15·21-s + 0.417i·23-s + (−0.600 − 0.800i)25-s − 2.78i·27-s − 0.371·29-s + 1.88·31-s + (−1.51 − 0.755i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.204861574\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.204861574\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 7 | \( 1 - 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 9.23iT - 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 8.47iT - 43T^{2} \) |
| 47 | \( 1 + 3.52iT - 47T^{2} \) |
| 53 | \( 1 + 6.76iT - 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 3.23iT - 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 2.47iT - 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 - 2.94iT - 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 18.1iT - 97T^{2} \) |
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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
−9.888421538521867233010809777206, −9.456351785851899762351117287251, −8.459153214883137314013548289220, −8.047489936523633317950057257069, −6.42881252601734286860390947471, −5.78904643638252697454710700331, −4.98754678882548136076249449768, −4.08466713788181593607442808809, −3.10673657472407400143653004563, −2.57464255619718665494147186818,
0.55281636755501439907646968997, 1.12207663111506141448434994609, 2.41357424528843728019050976638, 3.77424793644705896764644534687, 4.71061199632861557512404499118, 5.85924386847324735692133598094, 6.75748918684056387131139544478, 7.45964753402991633906607753505, 7.78292251149096120229408754649, 8.666312720823489493774794085946