L(s) = 1 | + (0.618 + 1.61i)3-s + 2.23·5-s + (1 + i)7-s + (−2.23 + 2.00i)9-s − 4.47i·11-s + (−3 + 3i)13-s + (1.38 + 3.61i)15-s + (2.23 − 2.23i)17-s + 2i·19-s + (−1 + 2.23i)21-s + (2.23 + 2.23i)23-s + 5.00·25-s + (−4.61 − 2.38i)27-s − 4.47·29-s − 4·31-s + ⋯ |
L(s) = 1 | + (0.356 + 0.934i)3-s + 0.999·5-s + (0.377 + 0.377i)7-s + (−0.745 + 0.666i)9-s − 1.34i·11-s + (−0.832 + 0.832i)13-s + (0.356 + 0.934i)15-s + (0.542 − 0.542i)17-s + 0.458i·19-s + (−0.218 + 0.487i)21-s + (0.466 + 0.466i)23-s + 1.00·25-s + (−0.888 − 0.458i)27-s − 0.830·29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40138 + 0.692953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40138 + 0.692953i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 - 1.61i)T \) |
| 5 | \( 1 - 2.23T \) |
good | 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.23 + 2.23i)T - 17iT^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + (-2.23 - 2.23i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.94iT - 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.70 + 6.70i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.23 - 2.23i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-1 - i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.47iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (6.70 + 6.70i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + (-9 - 9i)T + 97iT^{2} \) |
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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
−12.13115066161320219377902936827, −11.12988092235765997920627385626, −10.29585846296444545320743752773, −9.237980633035091110644984590471, −8.806836119434690603895943353598, −7.38320574069158203054610038036, −5.77208246777687827150158630480, −5.16813958803800113904499677103, −3.60271592086113855295189830632, −2.23804720887372384460904940647,
1.57657519551515762396399213495, 2.81366727372100398059723332847, 4.77540700702460406257143055254, 5.96885691089216655569722561515, 7.13510109905449442202161937971, 7.81342454328930457337331884930, 9.133371227550648904924832666472, 9.963716604656607285115406418372, 10.99467271289138221075067801193, 12.46180928359820454140786385982