L(s) = 1 | + (0.714 + 1.57i)3-s + (1.31 − 1.31i)5-s + 7-s + (−1.97 + 2.25i)9-s + (1.23 + 1.23i)11-s + (4.57 − 4.57i)13-s + (3.02 + 1.13i)15-s − 5.82i·17-s + (2.27 + 2.27i)19-s + (0.714 + 1.57i)21-s + 2.63i·23-s + 1.51i·25-s + (−4.97 − 1.50i)27-s + (−4.30 − 4.30i)29-s − 5.81i·31-s + ⋯ |
L(s) = 1 | + (0.412 + 0.910i)3-s + (0.590 − 0.590i)5-s + 0.377·7-s + (−0.659 + 0.751i)9-s + (0.372 + 0.372i)11-s + (1.26 − 1.26i)13-s + (0.781 + 0.294i)15-s − 1.41i·17-s + (0.521 + 0.521i)19-s + (0.155 + 0.344i)21-s + 0.549i·23-s + 0.303i·25-s + (−0.956 − 0.290i)27-s + (−0.799 − 0.799i)29-s − 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.378653886\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.378653886\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.714 - 1.57i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-1.31 + 1.31i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1.23 - 1.23i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.57 + 4.57i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.82iT - 17T^{2} \) |
| 19 | \( 1 + (-2.27 - 2.27i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.63iT - 23T^{2} \) |
| 29 | \( 1 + (4.30 + 4.30i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.81iT - 31T^{2} \) |
| 37 | \( 1 + (-6.99 - 6.99i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 + (2.12 - 2.12i)T - 43iT^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + (0.573 - 0.573i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.42 + 8.42i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.80 - 6.80i)T - 61iT^{2} \) |
| 67 | \( 1 + (1.90 + 1.90i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.3iT - 71T^{2} \) |
| 73 | \( 1 - 0.516iT - 73T^{2} \) |
| 79 | \( 1 - 3.10iT - 79T^{2} \) |
| 83 | \( 1 + (2.91 - 2.91i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 - 8.09T + 97T^{2} \) |
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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
−9.505529483500971301352727168915, −9.128278812641548723717189598937, −8.082181978532350289486318675549, −7.55783908741519542787811153242, −5.96210675910094265273574504867, −5.47544164388302632662680816466, −4.56193223340994384615255711834, −3.62513044151030503900418852645, −2.59688960766049518705622741033, −1.17638340628480182380190847331,
1.30431572016503203389411814410, 2.13032700548036175083650495496, 3.31754568932048796674170004434, 4.24545676611718685619875587225, 5.83268560587512213215018169546, 6.28702273834271983301879097841, 7.03145345442798325577495057489, 7.930365091958662491550647036603, 8.960477577643987172802427400908, 9.090685304137755551270286913563