| L(s) = 1 | − 2-s + 1.96·3-s + 4-s − 4.18·5-s − 1.96·6-s + 7-s − 8-s + 0.866·9-s + 4.18·10-s + 1.96·12-s + 4.40·13-s − 14-s − 8.22·15-s + 16-s − 1.10·17-s − 0.866·18-s + 1.43·19-s − 4.18·20-s + 1.96·21-s − 1.16·23-s − 1.96·24-s + 12.4·25-s − 4.40·26-s − 4.19·27-s + 28-s + 7.30·29-s + 8.22·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.13·3-s + 0.5·4-s − 1.87·5-s − 0.802·6-s + 0.377·7-s − 0.353·8-s + 0.288·9-s + 1.32·10-s + 0.567·12-s + 1.22·13-s − 0.267·14-s − 2.12·15-s + 0.250·16-s − 0.267·17-s − 0.204·18-s + 0.329·19-s − 0.935·20-s + 0.429·21-s − 0.243·23-s − 0.401·24-s + 2.49·25-s − 0.863·26-s − 0.807·27-s + 0.188·28-s + 1.35·29-s + 1.50·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315756071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315756071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 13 | \( 1 - 4.40T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 + 2.83T + 31T^{2} \) |
| 37 | \( 1 + 7.08T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 + 7.19T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 7.41T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 8.47T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 + 6.44T + 83T^{2} \) |
| 89 | \( 1 - 7.52T + 89T^{2} \) |
| 97 | \( 1 - 1.40T + 97T^{2} \) |
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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
−8.859370328477168613989763814152, −8.516155780017734521271168280965, −8.040398343989802433852070334457, −7.34055267963391146078704101234, −6.53709720013788042459201579795, −5.09982150003989031565779557627, −3.83909532620213455429559783133, −3.51184197675877666087188020847, −2.34209432174308768197694169142, −0.841080825211494277461808925690,
0.841080825211494277461808925690, 2.34209432174308768197694169142, 3.51184197675877666087188020847, 3.83909532620213455429559783133, 5.09982150003989031565779557627, 6.53709720013788042459201579795, 7.34055267963391146078704101234, 8.040398343989802433852070334457, 8.516155780017734521271168280965, 8.859370328477168613989763814152
