L(s) = 1 | − 2.60·3-s − i·5-s − 2.76i·7-s + 3.76·9-s − 2.16i·11-s + (−0.167 − 3.60i)13-s + 2.60i·15-s + 5.03i·19-s + 7.20i·21-s + 4.93·23-s − 25-s − 2.00·27-s − 4.43·29-s − 3.37i·31-s + 5.63i·33-s + ⋯ |
L(s) = 1 | − 1.50·3-s − 0.447i·5-s − 1.04i·7-s + 1.25·9-s − 0.653i·11-s + (−0.0463 − 0.998i)13-s + 0.671i·15-s + 1.15i·19-s + 1.57i·21-s + 1.02·23-s − 0.200·25-s − 0.384·27-s − 0.823·29-s − 0.605i·31-s + 0.981i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4192094316\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4192094316\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (0.167 + 3.60i)T \) |
good | 3 | \( 1 + 2.60T + 3T^{2} \) |
| 7 | \( 1 + 2.76iT - 7T^{2} \) |
| 11 | \( 1 + 2.16iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.03iT - 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 + 3.37iT - 31T^{2} \) |
| 37 | \( 1 + 3.97iT - 37T^{2} \) |
| 41 | \( 1 - 1.66iT - 41T^{2} \) |
| 43 | \( 1 + 9.80T + 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.10iT - 67T^{2} \) |
| 71 | \( 1 - 16.1iT - 71T^{2} \) |
| 73 | \( 1 - 15.9iT - 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 3.97iT - 83T^{2} \) |
| 89 | \( 1 + 7.94iT - 89T^{2} \) |
| 97 | \( 1 + 0.462iT - 97T^{2} \) |
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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
−9.897438732319476880749305858723, −8.627084910889388566118439330891, −7.70078450015180593900362688274, −6.89907174082525367433432522482, −5.87997788749276626716052060161, −5.37864346065838155300787139111, −4.38997021838414185833428904590, −3.37451461165066725031040200521, −1.31047403035555371636557785297, −0.25157750600542780009122957912,
1.70070502673456382220951555435, 3.02915261222446577767822506124, 4.66066292165129229588555299350, 5.08490838069994134685934006176, 6.23052091877607180543411718032, 6.66293220894506541000174710599, 7.56473397445938650442714885665, 8.954323569376293540260334886936, 9.498784694910238888547798161784, 10.57555254488736789778563551156