| L(s) = 1 | − 1.91·2-s − 3.61·3-s − 4.34·4-s − 15.8·5-s + 6.90·6-s + 7·7-s + 23.5·8-s − 13.9·9-s + 30.2·10-s − 28.2·11-s + 15.7·12-s + 57.6·13-s − 13.3·14-s + 57.2·15-s − 10.2·16-s + 26.6·18-s − 82.0·19-s + 68.9·20-s − 25.2·21-s + 53.9·22-s + 170.·23-s − 85.2·24-s + 126.·25-s − 110.·26-s + 147.·27-s − 30.4·28-s − 110.·29-s + ⋯ |
| L(s) = 1 | − 0.675·2-s − 0.695·3-s − 0.543·4-s − 1.41·5-s + 0.469·6-s + 0.377·7-s + 1.04·8-s − 0.516·9-s + 0.958·10-s − 0.773·11-s + 0.377·12-s + 1.22·13-s − 0.255·14-s + 0.986·15-s − 0.160·16-s + 0.349·18-s − 0.991·19-s + 0.771·20-s − 0.262·21-s + 0.522·22-s + 1.54·23-s − 0.724·24-s + 1.01·25-s − 0.830·26-s + 1.05·27-s − 0.205·28-s − 0.708·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2733649134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2733649134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.91T + 8T^{2} \) |
| 3 | \( 1 + 3.61T + 27T^{2} \) |
| 5 | \( 1 + 15.8T + 125T^{2} \) |
| 11 | \( 1 + 28.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.6T + 2.19e3T^{2} \) |
| 19 | \( 1 + 82.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 170.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 30.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 36.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 519.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 90.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 637.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 669.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 48.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 814.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 677.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 6.39T + 4.93e5T^{2} \) |
| 83 | \( 1 + 427.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 522.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 172.T + 9.12e5T^{2} \) |
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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
−8.685536179557053963523641815766, −8.063934582044232376901194450565, −7.56856545059873120309673030879, −6.51801464416618184637494128980, −5.50436370565553647380902605447, −4.71042242331773466070139601527, −4.01421497763879783785421754262, −2.99186804423733198803291235134, −1.31078514590526954416328623113, −0.31109721168633325520304565260,
0.31109721168633325520304565260, 1.31078514590526954416328623113, 2.99186804423733198803291235134, 4.01421497763879783785421754262, 4.71042242331773466070139601527, 5.50436370565553647380902605447, 6.51801464416618184637494128980, 7.56856545059873120309673030879, 8.063934582044232376901194450565, 8.685536179557053963523641815766
