| L(s) = 1 | − 2.02·2-s − 8.82·3-s − 3.90·4-s + 19.4·5-s + 17.8·6-s − 18.7·7-s + 24.0·8-s + 50.8·9-s − 39.4·10-s − 7.50·11-s + 34.4·12-s + 20.1·13-s + 38.0·14-s − 171.·15-s − 17.5·16-s − 73.2·17-s − 102.·18-s − 47.8·19-s − 76.0·20-s + 165.·21-s + 15.1·22-s + 0.911·23-s − 212.·24-s + 254.·25-s − 40.7·26-s − 210.·27-s + 73.2·28-s + ⋯ |
| L(s) = 1 | − 0.715·2-s − 1.69·3-s − 0.487·4-s + 1.74·5-s + 1.21·6-s − 1.01·7-s + 1.06·8-s + 1.88·9-s − 1.24·10-s − 0.205·11-s + 0.828·12-s + 0.429·13-s + 0.725·14-s − 2.95·15-s − 0.274·16-s − 1.04·17-s − 1.34·18-s − 0.577·19-s − 0.850·20-s + 1.72·21-s + 0.147·22-s + 0.00826·23-s − 1.80·24-s + 2.03·25-s − 0.307·26-s − 1.49·27-s + 0.494·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 + 2.02T + 8T^{2} \) |
| 3 | \( 1 + 8.82T + 27T^{2} \) |
| 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 + 18.7T + 343T^{2} \) |
| 11 | \( 1 + 7.50T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 47.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.911T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 211.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 334.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 179.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 333.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 599.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 325.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 263.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 182.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 74.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 794.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 19.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 129.T + 9.12e5T^{2} \) |
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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
−9.438602438683719985188535403785, −8.804798385253534560717277435296, −7.27402326183527541417445594958, −6.39813680095447055918942373232, −5.90659872282197938427073434627, −5.13748690419856222930693069535, −4.12555616517325365532450438426, −2.21899976834563412642540241651, −1.03960241309839689996575661042, 0,
1.03960241309839689996575661042, 2.21899976834563412642540241651, 4.12555616517325365532450438426, 5.13748690419856222930693069535, 5.90659872282197938427073434627, 6.39813680095447055918942373232, 7.27402326183527541417445594958, 8.804798385253534560717277435296, 9.438602438683719985188535403785
