L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 21.5·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 43.0·10-s − 33.5·11-s + 12·12-s − 46.6·13-s − 14·14-s + 64.5·15-s + 16·16-s − 41.8·17-s − 18·18-s − 19·19-s + 86.1·20-s + 21·21-s + 67.0·22-s + 52.5·23-s − 24·24-s + 338.·25-s + 93.3·26-s + 27·27-s + 28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.92·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.36·10-s − 0.918·11-s + 0.288·12-s − 0.996·13-s − 0.267·14-s + 1.11·15-s + 0.250·16-s − 0.597·17-s − 0.235·18-s − 0.229·19-s + 0.962·20-s + 0.218·21-s + 0.649·22-s + 0.476·23-s − 0.204·24-s + 2.70·25-s + 0.704·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.684726476\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.684726476\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 + 19T \) |
good | 5 | \( 1 - 21.5T + 125T^{2} \) |
| 11 | \( 1 + 33.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 52.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 240.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 3.25T + 6.89e4T^{2} \) |
| 43 | \( 1 - 486.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 176.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 751.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 306.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 39.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 582.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 586.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 759.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 300.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 826.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 455.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 528.T + 9.12e5T^{2} \) |
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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
−9.962291906364829906743264331424, −9.074229844172003532598177775645, −8.418597743550267119083914220460, −7.36302323682049974800606716779, −6.50887901282638452657954649451, −5.53688647781536747230479761819, −4.65372271397177601728398297227, −2.55247184454947843764340915107, −2.40577868878851121014792129784, −1.02734491058775931517426532333,
1.02734491058775931517426532333, 2.40577868878851121014792129784, 2.55247184454947843764340915107, 4.65372271397177601728398297227, 5.53688647781536747230479761819, 6.50887901282638452657954649451, 7.36302323682049974800606716779, 8.418597743550267119083914220460, 9.074229844172003532598177775645, 9.962291906364829906743264331424