| L(s) = 1 | − 8.63·3-s + 20.7·5-s − 0.399·7-s + 47.6·9-s − 0.469·11-s − 0.471·13-s − 179.·15-s + 17·17-s + 135.·19-s + 3.45·21-s + 40.3·23-s + 307.·25-s − 178.·27-s + 259.·29-s − 183.·31-s + 4.05·33-s − 8.30·35-s + 15.8·37-s + 4.07·39-s − 194.·41-s + 368.·43-s + 990.·45-s − 536.·47-s − 342.·49-s − 146.·51-s + 161.·53-s − 9.76·55-s + ⋯ |
| L(s) = 1 | − 1.66·3-s + 1.85·5-s − 0.0215·7-s + 1.76·9-s − 0.0128·11-s − 0.0100·13-s − 3.09·15-s + 0.242·17-s + 1.64·19-s + 0.0358·21-s + 0.365·23-s + 2.45·25-s − 1.26·27-s + 1.66·29-s − 1.06·31-s + 0.0214·33-s − 0.0401·35-s + 0.0705·37-s + 0.0167·39-s − 0.739·41-s + 1.30·43-s + 3.28·45-s − 1.66·47-s − 0.999·49-s − 0.403·51-s + 0.418·53-s − 0.0239·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.889260008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.889260008\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 + 8.63T + 27T^{2} \) |
| 5 | \( 1 - 20.7T + 125T^{2} \) |
| 7 | \( 1 + 0.399T + 343T^{2} \) |
| 11 | \( 1 + 0.469T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.471T + 2.19e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 40.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 15.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 536.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 161.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 645.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 76.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 606.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 8.25T + 3.57e5T^{2} \) |
| 73 | \( 1 - 98.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 509.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 466.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−9.749022748354097700965142772726, −9.020861057185796844486914341705, −7.53494582616011762178010830619, −6.54570171413792484388605950035, −6.09791137868116665601578380974, −5.23455380098699411276663774693, −4.87287904900708562735896063541, −3.08104281888826524820149296859, −1.68211944138909016699579079016, −0.841976856157492148025288902168,
0.841976856157492148025288902168, 1.68211944138909016699579079016, 3.08104281888826524820149296859, 4.87287904900708562735896063541, 5.23455380098699411276663774693, 6.09791137868116665601578380974, 6.54570171413792484388605950035, 7.53494582616011762178010830619, 9.020861057185796844486914341705, 9.749022748354097700965142772726
