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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7531061163\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7531061163\) |
\(L(1)\) |
|
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
−12.82692121528171929420929785154, −10.88776294237400181453972288037, −10.34988003730379481310408726141, −9.776444448401661752872732518355, −8.675369046753481445219357500126, −7.54135244622423929370694674942, −6.41855386499090721004024537448, −5.40567304635705398602777906994, −4.34726461970485868771277807788, −3.31518019814357706344161295430,
1.67322738982507795003024301670, 2.39305163480691963301849715955, 4.44019658256500854570704243426, 5.72664450290764753759245159447, 6.66647484461294080438301284079, 8.245557687611407096365883087303, 8.868116711051517590274478162019, 9.727025252438298025371408438012, 11.10812518653009356999056463614, 12.09425021325017863686906424066