| L(s) = 1 | − 3.27e4·2-s + 3.85e7·3-s + 1.07e9·4-s + 1.00e11·5-s − 1.26e12·6-s − 1.29e13·7-s − 3.51e13·8-s + 8.66e14·9-s − 3.29e15·10-s + 1.88e16·11-s + 4.13e16·12-s − 4.84e16·13-s + 4.23e17·14-s + 3.87e18·15-s + 1.15e18·16-s + 2.81e17·17-s − 2.83e19·18-s − 2.23e19·19-s + 1.08e20·20-s − 4.97e20·21-s − 6.18e20·22-s + 1.98e21·23-s − 1.35e21·24-s + 5.47e21·25-s + 1.58e21·26-s + 9.58e21·27-s − 1.38e22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.55·3-s + 0.5·4-s + 1.47·5-s − 1.09·6-s − 1.02·7-s − 0.353·8-s + 1.40·9-s − 1.04·10-s + 1.36·11-s + 0.775·12-s − 0.262·13-s + 0.726·14-s + 2.28·15-s + 0.250·16-s + 0.0238·17-s − 0.992·18-s − 0.337·19-s + 0.737·20-s − 1.59·21-s − 0.963·22-s + 1.55·23-s − 0.548·24-s + 1.17·25-s + 0.185·26-s + 0.624·27-s − 0.514·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(\approx\) |
\(2.569140336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.569140336\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3.27e4T \) |
| good | 3 | \( 1 - 3.85e7T + 6.17e14T^{2} \) |
| 5 | \( 1 - 1.00e11T + 4.65e21T^{2} \) |
| 7 | \( 1 + 1.29e13T + 1.57e26T^{2} \) |
| 11 | \( 1 - 1.88e16T + 1.91e32T^{2} \) |
| 13 | \( 1 + 4.84e16T + 3.40e34T^{2} \) |
| 17 | \( 1 - 2.81e17T + 1.39e38T^{2} \) |
| 19 | \( 1 + 2.23e19T + 4.37e39T^{2} \) |
| 23 | \( 1 - 1.98e21T + 1.63e42T^{2} \) |
| 29 | \( 1 - 7.83e21T + 2.15e45T^{2} \) |
| 31 | \( 1 + 7.33e22T + 1.70e46T^{2} \) |
| 37 | \( 1 - 2.48e24T + 4.11e48T^{2} \) |
| 41 | \( 1 + 1.78e25T + 9.91e49T^{2} \) |
| 43 | \( 1 + 2.70e25T + 4.34e50T^{2} \) |
| 47 | \( 1 - 3.94e25T + 6.83e51T^{2} \) |
| 53 | \( 1 + 7.36e26T + 2.83e53T^{2} \) |
| 59 | \( 1 + 7.94e26T + 7.87e54T^{2} \) |
| 61 | \( 1 - 1.03e27T + 2.21e55T^{2} \) |
| 67 | \( 1 - 1.29e27T + 4.05e56T^{2} \) |
| 71 | \( 1 + 2.64e28T + 2.44e57T^{2} \) |
| 73 | \( 1 + 5.13e28T + 5.79e57T^{2} \) |
| 79 | \( 1 + 2.35e29T + 6.70e58T^{2} \) |
| 83 | \( 1 - 4.15e29T + 3.10e59T^{2} \) |
| 89 | \( 1 - 1.32e29T + 2.69e60T^{2} \) |
| 97 | \( 1 - 7.28e30T + 3.88e61T^{2} \) |
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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
−20.20488582073873889568284906176, −18.97780315211315065396241743829, −16.98194262506662225952897443350, −14.70894182662024891153975298837, −13.24032858658847570835527694803, −9.768600610567284473766511484641, −8.981253192739081945156366818151, −6.64626188904998780763796389567, −3.06058363866767942588973868552, −1.62918593769605815388407559532,
1.62918593769605815388407559532, 3.06058363866767942588973868552, 6.64626188904998780763796389567, 8.981253192739081945156366818151, 9.768600610567284473766511484641, 13.24032858658847570835527694803, 14.70894182662024891153975298837, 16.98194262506662225952897443350, 18.97780315211315065396241743829, 20.20488582073873889568284906176
