| L(s) = 1 | + 1.25e6·2-s − 1.16e9·3-s + 1.02e12·4-s − 6.13e12·5-s − 1.45e15·6-s + 3.89e16·7-s + 5.89e17·8-s + 1.35e18·9-s − 7.68e18·10-s − 1.99e20·11-s − 1.18e21·12-s + 6.42e21·13-s + 4.87e22·14-s + 7.12e21·15-s + 1.77e23·16-s + 1.81e24·17-s + 1.69e24·18-s + 6.82e24·19-s − 6.25e24·20-s − 4.52e25·21-s − 2.49e26·22-s + 4.57e26·23-s − 6.84e26·24-s − 1.78e27·25-s + 8.04e27·26-s − 1.57e27·27-s + 3.97e28·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.85·4-s − 0.143·5-s − 0.975·6-s + 1.29·7-s + 1.44·8-s + 0.333·9-s − 0.242·10-s − 0.981·11-s − 1.07·12-s + 1.21·13-s + 2.18·14-s + 0.0829·15-s + 0.587·16-s + 1.84·17-s + 0.563·18-s + 0.791·19-s − 0.266·20-s − 0.745·21-s − 1.65·22-s + 1.27·23-s − 0.834·24-s − 0.979·25-s + 2.05·26-s − 0.192·27-s + 2.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(20)\) |
\(\approx\) |
\(5.209309315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.209309315\) |
| \(L(\frac{41}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.16e9T \) |
| good | 2 | \( 1 - 1.25e6T + 5.49e11T^{2} \) |
| 5 | \( 1 + 6.13e12T + 1.81e27T^{2} \) |
| 7 | \( 1 - 3.89e16T + 9.09e32T^{2} \) |
| 11 | \( 1 + 1.99e20T + 4.11e40T^{2} \) |
| 13 | \( 1 - 6.42e21T + 2.77e43T^{2} \) |
| 17 | \( 1 - 1.81e24T + 9.71e47T^{2} \) |
| 19 | \( 1 - 6.82e24T + 7.43e49T^{2} \) |
| 23 | \( 1 - 4.57e26T + 1.28e53T^{2} \) |
| 29 | \( 1 - 3.57e28T + 1.08e57T^{2} \) |
| 31 | \( 1 + 1.66e29T + 1.45e58T^{2} \) |
| 37 | \( 1 - 2.76e30T + 1.44e61T^{2} \) |
| 41 | \( 1 + 5.74e30T + 7.91e62T^{2} \) |
| 43 | \( 1 + 3.87e31T + 5.07e63T^{2} \) |
| 47 | \( 1 + 2.60e32T + 1.62e65T^{2} \) |
| 53 | \( 1 + 1.46e32T + 1.76e67T^{2} \) |
| 59 | \( 1 + 1.10e33T + 1.15e69T^{2} \) |
| 61 | \( 1 + 9.76e34T + 4.24e69T^{2} \) |
| 67 | \( 1 + 4.32e35T + 1.64e71T^{2} \) |
| 71 | \( 1 + 8.50e35T + 1.58e72T^{2} \) |
| 73 | \( 1 + 2.18e36T + 4.67e72T^{2} \) |
| 79 | \( 1 - 1.19e37T + 1.01e74T^{2} \) |
| 83 | \( 1 - 1.67e37T + 6.98e74T^{2} \) |
| 89 | \( 1 - 1.45e38T + 1.06e76T^{2} \) |
| 97 | \( 1 + 7.79e38T + 3.04e77T^{2} \) |
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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
−16.19118832955298008918628071607, −14.81522468805728535591554272751, −13.45959797543461921094243692178, −11.96525737344024458868288953903, −10.88450905266892113272519849528, −7.67905762148281441319403498354, −5.74503436072033728824460289475, −4.86601351642958178363589531726, −3.29631035790004653621081044578, −1.37779641069758828311165351227,
1.37779641069758828311165351227, 3.29631035790004653621081044578, 4.86601351642958178363589531726, 5.74503436072033728824460289475, 7.67905762148281441319403498354, 10.88450905266892113272519849528, 11.96525737344024458868288953903, 13.45959797543461921094243692178, 14.81522468805728535591554272751, 16.19118832955298008918628071607
