L(s) = 1 | − 1.41·3-s + (−1.41 + 1.73i)5-s + 2.44i·7-s − 0.999·9-s + 3.46i·11-s + (2.00 − 2.44i)15-s + 4.89i·17-s − 3.46i·19-s − 3.46i·21-s − 2.44i·23-s + (−0.999 − 4.89i)25-s + 5.65·27-s − 4·31-s − 4.89i·33-s + (−4.24 − 3.46i)35-s + ⋯ |
L(s) = 1 | − 0.816·3-s + (−0.632 + 0.774i)5-s + 0.925i·7-s − 0.333·9-s + 1.04i·11-s + (0.516 − 0.632i)15-s + 1.18i·17-s − 0.794i·19-s − 0.755i·21-s − 0.510i·23-s + (−0.199 − 0.979i)25-s + 1.08·27-s − 0.718·31-s − 0.852i·33-s + (−0.717 − 0.585i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.346515 + 0.502006i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.346515 + 0.502006i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.41 - 1.73i)T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 2.44iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 - 7.34iT - 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−12.82052026991340339886848972013, −12.10429844490402254867971559685, −11.24541033021269226051914186160, −10.48865480322677058745604468460, −9.142138410527000982977410231135, −7.923200270208475882931251061289, −6.69577593568696633425737797980, −5.75733877133451644015694259716, −4.37624647484079732059320414888, −2.61331280460558680389105991696,
0.64690881897807127230974538270, 3.55917925667897857737300198247, 4.90368684188128360402585046865, 5.95039729332544671843986894761, 7.35604329280083780527848613717, 8.361244593007290123258692281392, 9.559186991188726043214500081948, 10.94759574730425031054135082853, 11.46213209297231509743539821501, 12.42740691556158806841791236161