L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.830 − 0.830i)3-s − 1.00i·4-s + (2.05 − 0.880i)5-s + 1.17i·6-s + (0.707 + 0.707i)8-s + 1.62i·9-s + (−0.830 + 2.07i)10-s + 0.743·11-s + (−0.830 − 0.830i)12-s + (2.05 − 2.05i)13-s + (0.975 − 2.43i)15-s − 1.00·16-s + (4.63 + 4.63i)17-s + (−1.14 − 1.14i)18-s − 1.89·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.479 − 0.479i)3-s − 0.500i·4-s + (0.919 − 0.393i)5-s + 0.479i·6-s + (0.250 + 0.250i)8-s + 0.540i·9-s + (−0.262 + 0.656i)10-s + 0.224·11-s + (−0.239 − 0.239i)12-s + (0.570 − 0.570i)13-s + (0.251 − 0.629i)15-s − 0.250·16-s + (1.12 + 1.12i)17-s + (−0.270 − 0.270i)18-s − 0.434·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55665 - 0.0600140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55665 - 0.0600140i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.05 + 0.880i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.830 + 0.830i)T - 3iT^{2} \) |
| 11 | \( 1 - 0.743T + 11T^{2} \) |
| 13 | \( 1 + (-2.05 + 2.05i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.63 - 4.63i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 + (3.74 + 3.74i)T + 23iT^{2} \) |
| 29 | \( 1 + 9.69iT - 29T^{2} \) |
| 31 | \( 1 + 3.42iT - 31T^{2} \) |
| 37 | \( 1 + (1.88 - 1.88i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.817iT - 41T^{2} \) |
| 43 | \( 1 + (-1.59 - 1.59i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.33 - 3.33i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.52 - 3.52i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 - 6.07iT - 61T^{2} \) |
| 67 | \( 1 + (9.68 - 9.68i)T - 67iT^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + (6.25 - 6.25i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.58iT - 79T^{2} \) |
| 83 | \( 1 + (-9.23 + 9.23i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.03T + 89T^{2} \) |
| 97 | \( 1 + (-3.16 - 3.16i)T + 97iT^{2} \) |
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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
−10.45813056351398014478766236627, −10.13531472024032424133853469652, −8.915306847660097099349901071514, −8.251733732421509813408329418134, −7.57219460930687191086419729642, −6.18658112843330617298853759561, −5.73771119334757425710755415753, −4.29911301644184058718558399895, −2.50961151118831201339554291386, −1.34339764918394175443992410601,
1.51671869273612253336074160188, 2.95014803284386186120612938607, 3.78495125655794208406151908684, 5.27495645497366641480186769072, 6.46765813630085936897137231583, 7.38531426883934184857465095427, 8.761691541761391113157749934024, 9.237518518023646544024175012631, 10.00956376229518605109676770519, 10.69735291839616104149215428885