| L(s) = 1 | − 1.41·2-s + 4.41·5-s + 2.82·8-s − 6.24·10-s + 4.24·11-s + 13-s − 4.00·16-s − 1.41·17-s − 1.24·19-s − 6·22-s + 0.171·23-s + 14.4·25-s − 1.41·26-s − 5.82·29-s + 5.24·31-s + 2.00·34-s − 6.24·37-s + 1.75·38-s + 12.4·40-s + 3.17·41-s − 5·43-s − 0.242·46-s + 4.41·47-s − 20.4·50-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 1.97·5-s + 0.999·8-s − 1.97·10-s + 1.27·11-s + 0.277·13-s − 1.00·16-s − 0.342·17-s − 0.285·19-s − 1.27·22-s + 0.0357·23-s + 2.89·25-s − 0.277·26-s − 1.08·29-s + 0.941·31-s + 0.342·34-s − 1.02·37-s + 0.285·38-s + 1.97·40-s + 0.495·41-s − 0.762·43-s − 0.0357·46-s + 0.643·47-s − 2.89·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.848804238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.848804238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 - 4.41T + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 0.171T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 - 0.757T + 73T^{2} \) |
| 79 | \( 1 + 1.48T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.619649711018116961053442514040, −7.37792772773811150790819243122, −6.69720923938440194857575122868, −6.14436658816318858282892165430, −5.36603868276240398679002084714, −4.59054593982342223452032570314, −3.62865242360317512610247662570, −2.32301007185861096100691109910, −1.69359840170007934644986406203, −0.913565629397579784893698590861,
0.913565629397579784893698590861, 1.69359840170007934644986406203, 2.32301007185861096100691109910, 3.62865242360317512610247662570, 4.59054593982342223452032570314, 5.36603868276240398679002084714, 6.14436658816318858282892165430, 6.69720923938440194857575122868, 7.37792772773811150790819243122, 8.619649711018116961053442514040
