| L(s) = 1 | − 6.63·2-s + 12·4-s − 59.6·7-s + 132.·8-s − 252·11-s + 119.·13-s + 396·14-s − 1.26e3·16-s + 689.·17-s + 220·19-s + 1.67e3·22-s + 2.43e3·23-s − 792·26-s − 716.·28-s − 6.93e3·29-s + 6.75e3·31-s + 4.13e3·32-s − 4.57e3·34-s + 1.39e4·37-s − 1.45e3·38-s + 198·41-s + 417.·43-s − 3.02e3·44-s − 1.61e4·46-s + 1.05e4·47-s − 1.32e4·49-s + 1.43e3·52-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.375·4-s − 0.460·7-s + 0.732·8-s − 0.627·11-s + 0.195·13-s + 0.539·14-s − 1.23·16-s + 0.578·17-s + 0.139·19-s + 0.736·22-s + 0.959·23-s − 0.229·26-s − 0.172·28-s − 1.53·29-s + 1.26·31-s + 0.714·32-s − 0.678·34-s + 1.67·37-s − 0.163·38-s + 0.0183·41-s + 0.0344·43-s − 0.235·44-s − 1.12·46-s + 0.695·47-s − 0.787·49-s + 0.0734·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 6.63T + 32T^{2} \) |
| 7 | \( 1 + 59.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 252T + 1.61e5T^{2} \) |
| 13 | \( 1 - 119.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 689.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 220T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 198T + 1.15e8T^{2} \) |
| 43 | \( 1 - 417.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.82e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5T + 8.58e9T^{2} \) |
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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
−10.62441995446558768900099680517, −9.788242869420337006876097678141, −9.037649110502722844995713759263, −7.985718088998878583965738641626, −7.22106824889611419125741888276, −5.86013416013236333928805948267, −4.48529443556198924561564843327, −2.89023478441739077057282056318, −1.27520153306779139054072249281, 0,
1.27520153306779139054072249281, 2.89023478441739077057282056318, 4.48529443556198924561564843327, 5.86013416013236333928805948267, 7.22106824889611419125741888276, 7.985718088998878583965738641626, 9.037649110502722844995713759263, 9.788242869420337006876097678141, 10.62441995446558768900099680517
