L(s) = 1 | − 2·2-s + 4.98·3-s + 4·4-s + 2.34·5-s − 9.96·6-s + 2.84·7-s − 8·8-s − 2.16·9-s − 4.69·10-s + 3.80·11-s + 19.9·12-s − 16.3·13-s − 5.68·14-s + 11.7·15-s + 16·16-s − 74.7·17-s + 4.33·18-s + 9.39·20-s + 14.1·21-s − 7.60·22-s + 35.0·23-s − 39.8·24-s − 119.·25-s + 32.7·26-s − 145.·27-s + 11.3·28-s − 78.1·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.959·3-s + 0.5·4-s + 0.210·5-s − 0.678·6-s + 0.153·7-s − 0.353·8-s − 0.0802·9-s − 0.148·10-s + 0.104·11-s + 0.479·12-s − 0.349·13-s − 0.108·14-s + 0.201·15-s + 0.250·16-s − 1.06·17-s + 0.0567·18-s + 0.105·20-s + 0.147·21-s − 0.0736·22-s + 0.317·23-s − 0.339·24-s − 0.955·25-s + 0.247·26-s − 1.03·27-s + 0.0767·28-s − 0.500·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 4.98T + 27T^{2} \) |
| 5 | \( 1 - 2.34T + 125T^{2} \) |
| 7 | \( 1 - 2.84T + 343T^{2} \) |
| 11 | \( 1 - 3.80T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.7T + 4.91e3T^{2} \) |
| 23 | \( 1 - 35.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 78.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 401.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 64.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 77.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 465.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 70.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 663.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 157.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 89.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 631.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 758.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 824.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 132.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.32e3T + 9.12e5T^{2} \) |
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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
−9.475717082597421396312746486596, −8.670137319018558660151899398005, −8.132968303420826441318950002314, −7.16532082943349201281097988163, −6.26180458463314527773608260471, −5.03100369179258818583113075650, −3.69871834659420397138377399062, −2.61111817074683754028179570565, −1.72790112383009977982020326887, 0,
1.72790112383009977982020326887, 2.61111817074683754028179570565, 3.69871834659420397138377399062, 5.03100369179258818583113075650, 6.26180458463314527773608260471, 7.16532082943349201281097988163, 8.132968303420826441318950002314, 8.670137319018558660151899398005, 9.475717082597421396312746486596