L(s) = 1 | + (−1.63e3 + 1.63e3i)2-s − 3.24e6i·4-s + (5.62e5 + 2.18e7i)5-s + (3.62e8 + 3.62e8i)7-s + (1.86e9 + 1.86e9i)8-s + (−3.65e10 − 3.47e10i)10-s − 1.00e11i·11-s + (3.85e9 − 3.85e9i)13-s − 1.18e12·14-s + 6.93e11·16-s + (−8.99e12 + 8.99e12i)17-s − 3.16e13i·19-s + (7.07e13 − 1.82e12i)20-s + (1.63e14 + 1.63e14i)22-s + (1.75e14 + 1.75e14i)23-s + ⋯ |
L(s) = 1 | + (−1.12 + 1.12i)2-s − 1.54i·4-s + (0.0257 + 0.999i)5-s + (0.485 + 0.485i)7-s + (0.614 + 0.614i)8-s + (−1.15 − 1.09i)10-s − 1.16i·11-s + (0.00775 − 0.00775i)13-s − 1.09·14-s + 0.157·16-s + (−1.08 + 1.08i)17-s − 1.18i·19-s + (1.54 − 0.0398i)20-s + (1.31 + 1.31i)22-s + (0.883 + 0.883i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0361 - 0.999i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.0361 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(0.9483541297\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9483541297\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-5.62e5 - 2.18e7i)T \) |
good | 2 | \( 1 + (1.63e3 - 1.63e3i)T - 2.09e6iT^{2} \) |
| 7 | \( 1 + (-3.62e8 - 3.62e8i)T + 5.58e17iT^{2} \) |
| 11 | \( 1 + 1.00e11iT - 7.40e21T^{2} \) |
| 13 | \( 1 + (-3.85e9 + 3.85e9i)T - 2.47e23iT^{2} \) |
| 17 | \( 1 + (8.99e12 - 8.99e12i)T - 6.90e25iT^{2} \) |
| 19 | \( 1 + 3.16e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 + (-1.75e14 - 1.75e14i)T + 3.94e28iT^{2} \) |
| 29 | \( 1 - 1.14e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 6.89e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + (1.65e16 + 1.65e16i)T + 8.55e32iT^{2} \) |
| 41 | \( 1 - 2.61e16iT - 7.38e33T^{2} \) |
| 43 | \( 1 + (-4.43e16 + 4.43e16i)T - 2.00e34iT^{2} \) |
| 47 | \( 1 + (6.90e16 - 6.90e16i)T - 1.30e35iT^{2} \) |
| 53 | \( 1 + (3.02e17 + 3.02e17i)T + 1.62e36iT^{2} \) |
| 59 | \( 1 - 4.99e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 8.37e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + (-8.33e17 - 8.33e17i)T + 2.22e38iT^{2} \) |
| 71 | \( 1 + 3.96e19iT - 7.52e38T^{2} \) |
| 73 | \( 1 + (5.48e18 - 5.48e18i)T - 1.34e39iT^{2} \) |
| 79 | \( 1 + 1.46e20iT - 7.08e39T^{2} \) |
| 83 | \( 1 + (-6.35e19 - 6.35e19i)T + 1.99e40iT^{2} \) |
| 89 | \( 1 - 2.65e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + (-5.03e20 - 5.03e20i)T + 5.27e41iT^{2} \) |
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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
−11.33612473253125522546135082501, −10.63686357608316407574991339245, −9.172727706207426481533572600184, −8.448059978871575319388505504204, −7.26757415111667164995438507205, −6.39496775389322440801793364650, −5.39464867638840580102046001789, −3.43973601039400461863162431534, −1.98450135995784348423092709301, −0.50124979094673976203593152411,
0.59039937448873308842346508170, 1.49879987671959762573457001796, 2.37835848538027332113862321314, 4.01604241391013526453004454815, 5.09103048040951646235304820842, 7.15667256150252824668338338182, 8.304432083609582356935001896951, 9.204611375913485733672228328316, 10.10862580041939630755549178916, 11.19468928117421114462017787042