| L(s) = 1 | − 6.04e4·2-s + 1.43e7·3-s + 1.50e9·4-s − 5.19e10·5-s − 8.67e11·6-s + 4.79e11·7-s + 3.86e13·8-s + 2.05e14·9-s + 3.14e15·10-s + 2.28e16·11-s + 2.16e16·12-s − 1.55e17·13-s − 2.89e16·14-s − 7.45e17·15-s − 5.57e18·16-s − 8.05e17·17-s − 1.24e19·18-s + 1.89e19·19-s − 7.83e19·20-s + 6.87e18·21-s − 1.38e21·22-s − 2.34e21·23-s + 5.54e20·24-s − 1.95e21·25-s + 9.42e21·26-s + 2.95e21·27-s + 7.22e20·28-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.577·3-s + 0.702·4-s − 0.761·5-s − 0.753·6-s + 0.0381·7-s + 0.388·8-s + 0.333·9-s + 0.993·10-s + 1.65·11-s + 0.405·12-s − 0.844·13-s − 0.0497·14-s − 0.439·15-s − 1.20·16-s − 0.0682·17-s − 0.434·18-s + 0.286·19-s − 0.534·20-s + 0.0220·21-s − 2.15·22-s − 1.83·23-s + 0.224·24-s − 0.419·25-s + 1.10·26-s + 0.192·27-s + 0.0267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.43e7T \) |
| good | 2 | \( 1 + 6.04e4T + 2.14e9T^{2} \) |
| 5 | \( 1 + 5.19e10T + 4.65e21T^{2} \) |
| 7 | \( 1 - 4.79e11T + 1.57e26T^{2} \) |
| 11 | \( 1 - 2.28e16T + 1.91e32T^{2} \) |
| 13 | \( 1 + 1.55e17T + 3.40e34T^{2} \) |
| 17 | \( 1 + 8.05e17T + 1.39e38T^{2} \) |
| 19 | \( 1 - 1.89e19T + 4.37e39T^{2} \) |
| 23 | \( 1 + 2.34e21T + 1.63e42T^{2} \) |
| 29 | \( 1 - 4.26e22T + 2.15e45T^{2} \) |
| 31 | \( 1 + 2.06e23T + 1.70e46T^{2} \) |
| 37 | \( 1 + 3.53e24T + 4.11e48T^{2} \) |
| 41 | \( 1 + 2.76e24T + 9.91e49T^{2} \) |
| 43 | \( 1 - 1.70e25T + 4.34e50T^{2} \) |
| 47 | \( 1 + 9.50e25T + 6.83e51T^{2} \) |
| 53 | \( 1 - 6.00e26T + 2.83e53T^{2} \) |
| 59 | \( 1 - 2.97e27T + 7.87e54T^{2} \) |
| 61 | \( 1 - 3.11e27T + 2.21e55T^{2} \) |
| 67 | \( 1 + 7.95e27T + 4.05e56T^{2} \) |
| 71 | \( 1 + 5.35e28T + 2.44e57T^{2} \) |
| 73 | \( 1 + 1.27e28T + 5.79e57T^{2} \) |
| 79 | \( 1 + 3.59e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 4.19e29T + 3.10e59T^{2} \) |
| 89 | \( 1 + 1.62e30T + 2.69e60T^{2} \) |
| 97 | \( 1 - 6.75e30T + 3.88e61T^{2} \) |
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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
−17.67561900284002710512298734955, −16.15767680739050468337887796509, −14.31199601941628522589489653837, −11.82704705012575726832178691228, −9.842641148184267770987201065673, −8.528970930588172073533714884844, −7.18009746568930714560449307487, −3.99255245317134865122102547691, −1.68755407214605605303095610546, 0,
1.68755407214605605303095610546, 3.99255245317134865122102547691, 7.18009746568930714560449307487, 8.528970930588172073533714884844, 9.842641148184267770987201065673, 11.82704705012575726832178691228, 14.31199601941628522589489653837, 16.15767680739050468337887796509, 17.67561900284002710512298734955
