| L(s) = 1 | + 1.51·2-s − 5.69·4-s + 1.30·5-s − 3.41·7-s − 20.7·8-s + 1.98·10-s − 12.1·11-s + 42.4·13-s − 5.19·14-s + 13.9·16-s + 58.4·17-s − 36.2·19-s − 7.45·20-s − 18.5·22-s − 142.·23-s − 123.·25-s + 64.4·26-s + 19.4·28-s + 94.7·29-s + 304.·31-s + 187.·32-s + 88.8·34-s − 4.47·35-s + 29.1·37-s − 55.1·38-s − 27.2·40-s + 319.·41-s + ⋯ |
| L(s) = 1 | + 0.536·2-s − 0.711·4-s + 0.117·5-s − 0.184·7-s − 0.919·8-s + 0.0628·10-s − 0.334·11-s + 0.905·13-s − 0.0991·14-s + 0.218·16-s + 0.834·17-s − 0.438·19-s − 0.0833·20-s − 0.179·22-s − 1.29·23-s − 0.986·25-s + 0.486·26-s + 0.131·28-s + 0.606·29-s + 1.76·31-s + 1.03·32-s + 0.448·34-s − 0.0216·35-s + 0.129·37-s − 0.235·38-s − 0.107·40-s + 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 1.51T + 8T^{2} \) |
| 5 | \( 1 - 1.30T + 125T^{2} \) |
| 7 | \( 1 + 3.41T + 343T^{2} \) |
| 11 | \( 1 + 12.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 94.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 304.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 29.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 106.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 71.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 620.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 641.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 157.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 868.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 325.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 376.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 411.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 409.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 376.T + 9.12e5T^{2} \) |
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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
−8.122049920590670281537345433114, −7.919748291173465957314584768462, −6.28114772755107955022929962343, −6.09794678580434710302586963632, −5.06059318146366490769175748376, −4.24418603304559061636325673144, −3.53041424958576516529882974238, −2.56850831403179906568267816991, −1.17502836890382335894897209809, 0,
1.17502836890382335894897209809, 2.56850831403179906568267816991, 3.53041424958576516529882974238, 4.24418603304559061636325673144, 5.06059318146366490769175748376, 6.09794678580434710302586963632, 6.28114772755107955022929962343, 7.919748291173465957314584768462, 8.122049920590670281537345433114
