| L(s) = 1 | − 2.01·2-s + 7.09·3-s − 3.92·4-s − 5·5-s − 14.3·6-s − 8.46·7-s + 24.0·8-s + 23.3·9-s + 10.0·10-s − 11·11-s − 27.8·12-s − 33.6·13-s + 17.1·14-s − 35.4·15-s − 17.2·16-s + 98.1·17-s − 47.1·18-s + 19·19-s + 19.6·20-s − 60.0·21-s + 22.2·22-s + 90.8·23-s + 170.·24-s + 25·25-s + 68.0·26-s − 25.7·27-s + 33.1·28-s + ⋯ |
| L(s) = 1 | − 0.714·2-s + 1.36·3-s − 0.490·4-s − 0.447·5-s − 0.975·6-s − 0.457·7-s + 1.06·8-s + 0.865·9-s + 0.319·10-s − 0.301·11-s − 0.669·12-s − 0.718·13-s + 0.326·14-s − 0.610·15-s − 0.269·16-s + 1.40·17-s − 0.617·18-s + 0.229·19-s + 0.219·20-s − 0.624·21-s + 0.215·22-s + 0.823·23-s + 1.45·24-s + 0.200·25-s + 0.512·26-s − 0.183·27-s + 0.223·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 2.01T + 8T^{2} \) |
| 3 | \( 1 - 7.09T + 27T^{2} \) |
| 7 | \( 1 + 8.46T + 343T^{2} \) |
| 13 | \( 1 + 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 98.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 90.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 142.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 66.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 501.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 249.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 252.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 669.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 648.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 773.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 685.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 734.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.00e3T + 9.12e5T^{2} \) |
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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
−9.218448956760441768004200613846, −8.308958821555607930530154695340, −7.69949107975551586932748130981, −7.20664800017634764562210059358, −5.60644634407256077593023519530, −4.52470567965678972228117529282, −3.51801503832434085430707861442, −2.75899562036974395563209829934, −1.36116192206601786203013131454, 0,
1.36116192206601786203013131454, 2.75899562036974395563209829934, 3.51801503832434085430707861442, 4.52470567965678972228117529282, 5.60644634407256077593023519530, 7.20664800017634764562210059358, 7.69949107975551586932748130981, 8.308958821555607930530154695340, 9.218448956760441768004200613846
