L(s) = 1 | − 1.52·2-s − 3-s + 0.319·4-s + 3.31·5-s + 1.52·6-s − 2.60·7-s + 2.55·8-s + 9-s − 5.05·10-s + 0.140·11-s − 0.319·12-s − 13-s + 3.97·14-s − 3.31·15-s − 4.53·16-s − 2.15·17-s − 1.52·18-s + 4.38·19-s + 1.05·20-s + 2.60·21-s − 0.214·22-s − 2.12·23-s − 2.55·24-s + 6.00·25-s + 1.52·26-s − 27-s − 0.833·28-s + ⋯ |
L(s) = 1 | − 1.07·2-s − 0.577·3-s + 0.159·4-s + 1.48·5-s + 0.621·6-s − 0.986·7-s + 0.904·8-s + 0.333·9-s − 1.59·10-s + 0.0423·11-s − 0.0921·12-s − 0.277·13-s + 1.06·14-s − 0.856·15-s − 1.13·16-s − 0.523·17-s − 0.358·18-s + 1.00·19-s + 0.236·20-s + 0.569·21-s − 0.0456·22-s − 0.442·23-s − 0.522·24-s + 1.20·25-s + 0.298·26-s − 0.192·27-s − 0.157·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8223540170\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8223540170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 - 0.140T + 11T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 - 4.38T + 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 + 2.28T + 41T^{2} \) |
| 43 | \( 1 + 0.693T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 + 7.96T + 59T^{2} \) |
| 61 | \( 1 + 0.855T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 3.55T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 0.381T + 83T^{2} \) |
| 89 | \( 1 - 3.05T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.715573289527417359889040556400, −7.73000636337448175495262885576, −6.93993327263684540178083406536, −6.35825255815963639587530475599, −5.59921204930299540754792599207, −4.93354652748176824074227401484, −3.83992300333750789099628874173, −2.58768780317264639181225220958, −1.71176142290268763950917555167, −0.64168133391442753774688476664,
0.64168133391442753774688476664, 1.71176142290268763950917555167, 2.58768780317264639181225220958, 3.83992300333750789099628874173, 4.93354652748176824074227401484, 5.59921204930299540754792599207, 6.35825255815963639587530475599, 6.93993327263684540178083406536, 7.73000636337448175495262885576, 8.715573289527417359889040556400