L(s) = 1 | + i·2-s − 1.41i·3-s − 4-s + (−0.707 − 0.707i)5-s + 1.41·6-s − i·7-s − i·8-s − 1.00·9-s + (0.707 − 0.707i)10-s + 1.41i·12-s + 1.41i·13-s + 14-s + (−1.00 + 1.00i)15-s + 16-s − 1.00i·18-s + 1.41·19-s + ⋯ |
L(s) = 1 | + i·2-s − 1.41i·3-s − 4-s + (−0.707 − 0.707i)5-s + 1.41·6-s − i·7-s − i·8-s − 1.00·9-s + (0.707 − 0.707i)10-s + 1.41i·12-s + 1.41i·13-s + 14-s + (−1.00 + 1.00i)15-s + 16-s − 1.00i·18-s + 1.41·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6259243436\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6259243436\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−12.20898618630272905875130760286, −11.39076040648746491921282885073, −9.736460018491708411742477094034, −8.727191755292717787655958912301, −7.70262059668064359062657271905, −7.25753457813564924206943884245, −6.38754131075484860594574741791, −4.95575255671963414409236138270, −3.83008646919882056223614640972, −1.19131665193209874420607157325,
2.87554991319898098850264247325, 3.53872943663913842518785477498, 4.83123021718559643282327601472, 5.70606865385969620320556948495, 7.73410312088666485613325094090, 8.731615248253548933574620284630, 9.698222947804917371484020649155, 10.35542758146941242718469713287, 11.17899303870328303088224976438, 11.87632174711782255641358370677