| L(s) = 1 | − 2·2-s − 2.73·3-s + 4·4-s + 5.47·6-s − 3.11·7-s − 8·8-s − 19.5·9-s + 30.7·11-s − 10.9·12-s + 46.0·13-s + 6.23·14-s + 16·16-s + 65.4·17-s + 39.0·18-s − 145.·19-s + 8.53·21-s − 61.5·22-s − 23·23-s + 21.9·24-s − 92.0·26-s + 127.·27-s − 12.4·28-s + 121.·29-s − 112.·31-s − 32·32-s − 84.3·33-s − 130.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.527·3-s + 0.5·4-s + 0.372·6-s − 0.168·7-s − 0.353·8-s − 0.722·9-s + 0.844·11-s − 0.263·12-s + 0.981·13-s + 0.119·14-s + 0.250·16-s + 0.933·17-s + 0.510·18-s − 1.75·19-s + 0.0887·21-s − 0.596·22-s − 0.208·23-s + 0.186·24-s − 0.694·26-s + 0.907·27-s − 0.0841·28-s + 0.778·29-s − 0.652·31-s − 0.176·32-s − 0.444·33-s − 0.659·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9897518113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9897518113\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 2.73T + 27T^{2} \) |
| 7 | \( 1 + 3.11T + 343T^{2} \) |
| 11 | \( 1 - 30.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 145.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 112.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 346.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 573.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 241.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 846.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 82.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 178.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 238.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 450.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−9.401488394237259785224669293292, −8.522853925747844933399570294309, −8.087844818410080927892091861632, −6.67423441103197915666468460668, −6.31863892599810275726177589329, −5.40505950433151341447704089418, −4.10967081525412864880384312603, −3.08931096678818842822487063905, −1.74609593327426130619321526409, −0.59436912558734041442478623375,
0.59436912558734041442478623375, 1.74609593327426130619321526409, 3.08931096678818842822487063905, 4.10967081525412864880384312603, 5.40505950433151341447704089418, 6.31863892599810275726177589329, 6.67423441103197915666468460668, 8.087844818410080927892091861632, 8.522853925747844933399570294309, 9.401488394237259785224669293292
