L(s) = 1 | + (0.707 − 0.707i)3-s + (−2 − i)5-s + (−2.82 − 2.82i)7-s − 1.00i·9-s − 5.65i·11-s + (3 + 3i)13-s + (−2.12 + 0.707i)15-s + (−1 + i)17-s + 5.65·19-s − 4.00·21-s + (−2.82 + 2.82i)23-s + (3 + 4i)25-s + (−0.707 − 0.707i)27-s − 4i·29-s + (−4.00 − 4.00i)33-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.894 − 0.447i)5-s + (−1.06 − 1.06i)7-s − 0.333i·9-s − 1.70i·11-s + (0.832 + 0.832i)13-s + (−0.547 + 0.182i)15-s + (−0.242 + 0.242i)17-s + 1.29·19-s − 0.872·21-s + (−0.589 + 0.589i)23-s + (0.600 + 0.800i)25-s + (−0.136 − 0.136i)27-s − 0.742i·29-s + (−0.696 − 0.696i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.632971 - 0.799794i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.632971 - 0.799794i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (2 + i)T \) |
good | 7 | \( 1 + (2.82 + 2.82i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-5 + 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (2.82 - 2.82i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.82 + 2.82i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + 53iT^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 + (2.82 - 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
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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
−11.72651841409739509269833564717, −11.15442118608684676494326521269, −9.805536341914023380516093492183, −8.802082086202481331889449434264, −7.929585879600716202321505712217, −6.94004240702190873337035002937, −5.88407806821299090770941220007, −3.97312511185145943903058572119, −3.35734125906545015755668538688, −0.823085990101948667183261951531,
2.65260589556785103354728118126, 3.65979733201586147774947981284, 5.03932347281880045412044741604, 6.44721319712764612476562636391, 7.50219703658001508675906791419, 8.547337365992140253781221448752, 9.597360053269691218187965533859, 10.28325304748005029938055318529, 11.57136967262389364183438643461, 12.37674511076148931428553301701