| L(s) = 1 | + (−1 − i)3-s + (1 + 2i)5-s − i·9-s + (3 + 3i)11-s + (3 + 3i)13-s + (1 − 3i)15-s − 4i·17-s + (1 − i)19-s + 8·23-s + (−3 + 4i)25-s + (−4 + 4i)27-s + (3 − 3i)29-s − 6i·33-s + (3 − 3i)37-s − 6i·39-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.577i)3-s + (0.447 + 0.894i)5-s − 0.333i·9-s + (0.904 + 0.904i)11-s + (0.832 + 0.832i)13-s + (0.258 − 0.774i)15-s − 0.970i·17-s + (0.229 − 0.229i)19-s + 1.66·23-s + (−0.600 + 0.800i)25-s + (−0.769 + 0.769i)27-s + (0.557 − 0.557i)29-s − 1.04i·33-s + (0.493 − 0.493i)37-s − 0.960i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25671 + 0.0445996i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25671 + 0.0445996i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + (-3 - 3i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + (-1 + i)T - 19iT^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + (-3 + 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + (9 - 9i)T - 53iT^{2} \) |
| 59 | \( 1 + (9 + 9i)T + 59iT^{2} \) |
| 61 | \( 1 + (5 - 5i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (9 + 9i)T + 83iT^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−11.50215152050005401709128680070, −11.05951390877709589172759414747, −9.643367671978985906779767123385, −9.127905672266235030647007906370, −7.39783820172731734990049752671, −6.72399724381487767526207173912, −6.10328301280886839159462132034, −4.61808802857359198520093291147, −3.12641082032681766964758731637, −1.48006086817518078236915258391,
1.25589686520630742536294150701, 3.42622832915344619222425503842, 4.71413021304624055773597026749, 5.60110123390204556726549497183, 6.41768309490026811865075907519, 8.133244739273812371385877596602, 8.789685722396959186889625466541, 9.835349881938172990073592918218, 10.78648730050423430781798730642, 11.40132360649148055994037119652
