| L(s) = 1 | + (−1 − 2i)5-s + (−1 − i)7-s − 6i·11-s + (−1 − i)13-s + (−1 + i)17-s + 4·19-s + (−5 + 5i)23-s + (−3 + 4i)25-s + 8i·29-s − 2i·31-s + (−1 + 3i)35-s + (−5 + 5i)37-s − 6·41-s + (3 − 3i)43-s + (−7 − 7i)47-s + ⋯ |
| L(s) = 1 | + (−0.447 − 0.894i)5-s + (−0.377 − 0.377i)7-s − 1.80i·11-s + (−0.277 − 0.277i)13-s + (−0.242 + 0.242i)17-s + 0.917·19-s + (−1.04 + 1.04i)23-s + (−0.600 + 0.800i)25-s + 1.48i·29-s − 0.359i·31-s + (−0.169 + 0.507i)35-s + (−0.821 + 0.821i)37-s − 0.937·41-s + (0.457 − 0.457i)43-s + (−1.02 − 1.02i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5432764664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5432764664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (5 - 5i)T - 23iT^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (7 + 7i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (7 + 7i)T + 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (-9 - 9i)T + 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 + 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.982057308331074596731283697574, −8.364648117526184691794388783482, −7.66936865878311624725629441079, −6.67619422419343276746333024759, −5.63581219000486920348424013005, −5.07201701853113120937285396111, −3.72118649948981497388303662435, −3.27878314372279031627482886553, −1.46011209873107107542489244705, −0.21634035022507637690139021622,
2.02072749519473538072721194102, 2.83197027244221292361290427993, 4.04282229855996479363070751214, 4.76682241780314307731601842257, 6.01032625509004315312111165013, 6.80957139102874314198658143881, 7.42440428137278816054368060778, 8.173110521573455575441686732743, 9.406806471027853192097028606626, 9.869073141952054986676514834988
