| L(s) = 1 | − 1.36·2-s + 9.26·3-s − 6.14·4-s + 1.21·5-s − 12.6·6-s + 10.5·7-s + 19.2·8-s + 58.8·9-s − 1.65·10-s + 39.9·11-s − 56.8·12-s − 52.4·13-s − 14.3·14-s + 11.2·15-s + 22.8·16-s + 1.95·17-s − 80.1·18-s − 86.6·19-s − 7.46·20-s + 97.5·21-s − 54.4·22-s − 205.·23-s + 178.·24-s − 123.·25-s + 71.5·26-s + 294.·27-s − 64.6·28-s + ⋯ |
| L(s) = 1 | − 0.482·2-s + 1.78·3-s − 0.767·4-s + 0.108·5-s − 0.859·6-s + 0.568·7-s + 0.852·8-s + 2.17·9-s − 0.0524·10-s + 1.09·11-s − 1.36·12-s − 1.11·13-s − 0.274·14-s + 0.193·15-s + 0.356·16-s + 0.0279·17-s − 1.05·18-s − 1.04·19-s − 0.0834·20-s + 1.01·21-s − 0.527·22-s − 1.86·23-s + 1.51·24-s − 0.988·25-s + 0.539·26-s + 2.10·27-s − 0.436·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.36T + 8T^{2} \) |
| 3 | \( 1 - 9.26T + 27T^{2} \) |
| 5 | \( 1 - 1.21T + 125T^{2} \) |
| 7 | \( 1 - 10.5T + 343T^{2} \) |
| 11 | \( 1 - 39.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.95T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 38.7T + 6.89e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 532.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 71.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 1.90T + 2.26e5T^{2} \) |
| 67 | \( 1 + 925.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 877.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 661.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 103.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.58e3T + 9.12e5T^{2} \) |
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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
−8.536578583074325481365090377631, −7.85133795958354432395992304167, −7.50142948947314937068501062770, −6.26962279910958592059860064850, −4.90443618862196994913505640199, −4.08353258776166594580134984294, −3.58455977649279581751947105534, −2.05981305008075853992787689298, −1.71105472480676134513724550717, 0,
1.71105472480676134513724550717, 2.05981305008075853992787689298, 3.58455977649279581751947105534, 4.08353258776166594580134984294, 4.90443618862196994913505640199, 6.26962279910958592059860064850, 7.50142948947314937068501062770, 7.85133795958354432395992304167, 8.536578583074325481365090377631
