| L(s) = 1 | + 2.01·2-s − 3.32·3-s + 2.06·4-s − 3.71·5-s − 6.70·6-s − 2.20·7-s + 0.126·8-s + 8.04·9-s − 7.48·10-s + 2.41·11-s − 6.85·12-s + 0.458·13-s − 4.44·14-s + 12.3·15-s − 3.87·16-s − 17-s + 16.2·18-s + 2.28·19-s − 7.66·20-s + 7.33·21-s + 4.86·22-s − 0.421·24-s + 8.78·25-s + 0.924·26-s − 16.7·27-s − 4.55·28-s − 9.35·29-s + ⋯ |
| L(s) = 1 | + 1.42·2-s − 1.91·3-s + 1.03·4-s − 1.66·5-s − 2.73·6-s − 0.834·7-s + 0.0448·8-s + 2.68·9-s − 2.36·10-s + 0.728·11-s − 1.97·12-s + 0.127·13-s − 1.18·14-s + 3.18·15-s − 0.967·16-s − 0.242·17-s + 3.82·18-s + 0.524·19-s − 1.71·20-s + 1.60·21-s + 1.03·22-s − 0.0861·24-s + 1.75·25-s + 0.181·26-s − 3.23·27-s − 0.860·28-s − 1.73·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8993 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8993 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3569307964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3569307964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.01T + 2T^{2} \) |
| 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 - 0.458T + 13T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 29 | \( 1 + 9.35T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 1.91T + 53T^{2} \) |
| 59 | \( 1 + 9.64T + 59T^{2} \) |
| 61 | \( 1 + 2.29T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 0.879T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 5.94T + 89T^{2} \) |
| 97 | \( 1 + 6.17T + 97T^{2} \) |
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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
−7.23225823143603423666883580397, −6.73137859849110136992298012152, −6.34087689047099470431365622386, −5.50755374044628646591092274814, −4.96442892404005688764557757340, −4.33088135473874843437144059599, −3.66699386155912307175729651940, −3.33935377553719596863061129294, −1.61354665566879521882012102648, −0.25651865836000359332567463475,
0.25651865836000359332567463475, 1.61354665566879521882012102648, 3.33935377553719596863061129294, 3.66699386155912307175729651940, 4.33088135473874843437144059599, 4.96442892404005688764557757340, 5.50755374044628646591092274814, 6.34087689047099470431365622386, 6.73137859849110136992298012152, 7.23225823143603423666883580397
