L(s) = 1 | + (1.93 + 1.93i)3-s + (1.73 − 1.73i)5-s + 1.03i·7-s + 4.46i·9-s + (0.896 − 0.896i)11-s + (−3.73 − 3.73i)13-s + 6.69·15-s − 3.46·17-s + (−0.896 − 0.896i)19-s + (−1.99 + 1.99i)21-s + 6.69i·23-s − 0.999i·25-s + (−2.82 + 2.82i)27-s + (1.73 + 1.73i)29-s − 5.65·31-s + ⋯ |
L(s) = 1 | + (1.11 + 1.11i)3-s + (0.774 − 0.774i)5-s + 0.391i·7-s + 1.48i·9-s + (0.270 − 0.270i)11-s + (−1.03 − 1.03i)13-s + 1.72·15-s − 0.840·17-s + (−0.205 − 0.205i)19-s + (−0.436 + 0.436i)21-s + 1.39i·23-s − 0.199i·25-s + (−0.544 + 0.544i)27-s + (0.321 + 0.321i)29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75809 + 0.596793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75809 + 0.596793i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-1.93 - 1.93i)T + 3iT^{2} \) |
| 5 | \( 1 + (-1.73 + 1.73i)T - 5iT^{2} \) |
| 7 | \( 1 - 1.03iT - 7T^{2} \) |
| 11 | \( 1 + (-0.896 + 0.896i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.73 + 3.73i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + (0.896 + 0.896i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.69iT - 23T^{2} \) |
| 29 | \( 1 + (-1.73 - 1.73i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + (-0.267 + 0.267i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (-5.79 + 5.79i)T - 43iT^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + (-4.26 + 4.26i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.58 - 7.58i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.267 - 0.267i)T + 61iT^{2} \) |
| 67 | \( 1 + (2.96 + 2.96i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.69iT - 71T^{2} \) |
| 73 | \( 1 + 9.46iT - 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + (5.79 + 5.79i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.46iT - 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−12.26541005371930056177527321477, −10.86210927199404568926510662243, −9.915679864526637141091218359651, −9.178471912212720684099486029482, −8.687746076229629896044430575585, −7.45401611761923354667222081065, −5.65051792095341727937243508453, −4.84352508277310901171769690462, −3.51149908649555035306620279972, −2.22751409905012043109948860848,
1.92363720144388450694252278766, 2.73109019078727698079115608358, 4.40742518223259501489448033198, 6.42816171913845783294230460045, 6.88294011797325370910495329732, 7.87951052396593558132910707086, 9.006356631140951349996851290100, 9.811262094479631623076389723340, 10.93291999202350371988562216462, 12.19148418825548488192470325191