| L(s) = 1 | − 1.16·2-s − 8.58·3-s − 6.64·4-s − 19.8·5-s + 9.98·6-s + 5.27·7-s + 17.0·8-s + 46.6·9-s + 23.1·10-s − 48.8·11-s + 57.0·12-s + 0.649·13-s − 6.13·14-s + 170.·15-s + 33.3·16-s + 88.0·17-s − 54.2·18-s − 83.7·19-s + 132.·20-s − 45.2·21-s + 56.7·22-s + 7.35·23-s − 146.·24-s + 270.·25-s − 0.755·26-s − 168.·27-s − 35.0·28-s + ⋯ |
| L(s) = 1 | − 0.411·2-s − 1.65·3-s − 0.830·4-s − 1.77·5-s + 0.679·6-s + 0.284·7-s + 0.753·8-s + 1.72·9-s + 0.731·10-s − 1.33·11-s + 1.37·12-s + 0.0138·13-s − 0.117·14-s + 2.93·15-s + 0.520·16-s + 1.25·17-s − 0.710·18-s − 1.01·19-s + 1.47·20-s − 0.470·21-s + 0.550·22-s + 0.0666·23-s − 1.24·24-s + 2.16·25-s − 0.00569·26-s − 1.20·27-s − 0.236·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1814587971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1814587971\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.16T + 8T^{2} \) |
| 3 | \( 1 + 8.58T + 27T^{2} \) |
| 5 | \( 1 + 19.8T + 125T^{2} \) |
| 7 | \( 1 - 5.27T + 343T^{2} \) |
| 11 | \( 1 + 48.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.649T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 83.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.35T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 211.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 252.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 448.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 648.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 187.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 416.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 383.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 803.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.16e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 200.T + 9.12e5T^{2} \) |
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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
−8.666975719322018106025449374133, −7.920816097996893728601834092406, −7.59789496344104428335724018327, −6.52027321051258382662405349915, −5.47063456463143650526997168435, −4.72958294851717999862135692090, −4.34426717818811428013791054190, −3.16296568885885610108894799782, −1.11767854931790783737172023566, −0.28985580607289637040925613143,
0.28985580607289637040925613143, 1.11767854931790783737172023566, 3.16296568885885610108894799782, 4.34426717818811428013791054190, 4.72958294851717999862135692090, 5.47063456463143650526997168435, 6.52027321051258382662405349915, 7.59789496344104428335724018327, 7.920816097996893728601834092406, 8.666975719322018106025449374133
